There is a surprisingly modern collective response—or rather, range of responses—to this in ancient times. The 3 options listed below define a spectrum which could be described as moving from least to medium to most "in-universe."
1. Those myths and legends featuring mind-blowingly extraordinary stuff, such as the gods' trysts with humans, are just fictitious yarns with which the poets think to amuse themselves.
Much like at present day, there were philosophers who, perceiving themselves as more learned than the generally more illiterate populace, did not much share the classical mythographers' enthusiasm regarding the stories of rather human-like deities colliding with monsters and heroes in the realms of ordinary mortals.
In Miracle and Natural Law in Graeco-Roman and Early Christian Thought, Robert M. Grant1 lays out Lucian of Samosata's position on the matter (on pp. 71-72).
In his Lover of Lies Lucian vigorously attacks the "deceivers
of antiquity"—Herodotus, Ctesias (the author of the fantastic
Persica), and the poets, including Homer. They tell such stories as the castration of Uranus (such stories were often under attack), the
bonds of the giants, the metamorphoses of Zeus, and so on. These are
"dreadful and portentous myths". In opposition to stories of divine
healings Lucian claims that those who tell them and defend them are
reasoning from false premises; they "drive in a nail with a nail";
they cannot prove the presence of divine activity. To this objection
his opponents reply that disbelief in their stories implies disbelief
in the gods. Lucian answers that on the contrary he worships the gods
and takes note of their healings, which are actually effected by
medicines and physicians. Asclepius himself healed men in the same
way.
Grant is careful to note, however, that in Lucian's time, the 100s AD, this writer was unique for having a perspective as stark as this. For a philosophical viewpoint in this same vein, posited centuries before Lucian, see the section Centaurs
below.
2. There is a kernel of truth in these otherwise incredible fictions, but the real deal needs to be filtered out of the silliness surrounding this beeswax. Some of these things happened and some of these creatures existed but not necessarily in the manner related to us by those mythographers.
Oh, and people these days are rascals and reprobates [see more below]!
In the Description of Greece 8.2.4-7, Pausanias offers his opinion on the issue by way of his visit to Arkadia [Arcadia]. Lykaon [Lycaon], one of Arkadia's earliest kings, is supposed to have been turned into a wolf by his own grandfather, none other than Zeus, after the god had visited his house for a meal:
For the men of those days, because of their righteousness and
piety, were guests of the gods, eating at the same board; the good
were openly honored by the gods, and sinners were openly visited with
their wrath. As a matter of fact, in those days men were changed to
gods, who down to the present day have honours paid to them—Aristaios
[Aristaeus], Britomartis of Crete, Herakles [Heracles] the son of Alkmene,
Amphiaraos the son of Oikles, and besides these Polydeukes
[Polydeuces] and Kastor [Castor].
So one might believe that Lykaon was turned into a
beast, and Niobe, the daughter of Tantalos, into a stone. But at the
present time, when sin has grown to such a height and has been
spreading over every land and every city, no longer do men turn into
gods, except in the flattering words addressed to despots, and the
wrath of the gods is reserved until the sinners have departed to the
next world.
All through the ages, many events that have occurred in the past,
and even some that occur today, have been generally discredited
because of the lies built up on a foundation of fact. It is said, for
instance, that ever since the time of Lykaon a man has changed into a
wolf at the sacrifice to Zeus Lykaios [Lycaeus], but that the change
is not for life; if, when he is a wolf, he abstains from human flesh,
after nine years he becomes a man again, but if he tastes human flesh
he remains a beast for ever.
Similarly too it is said that Niobe on Mount Sipylos sheds tears
in the season of summer. I have also heard that the griffins have
spots like the leopard, and that the Tritons speak with human voice,
though others say that they blow through a shell that has been bored.
Those who like to listen to the miraculous are themselves apt to add
to the marvel, and so they ruin truth by mixing it with
falsehood.
In his poem Phainomena, from four hundred years prior to Pausanias, Aratus expands on this idea of human wickedness resulting in the disappearance of the gods. This divine departure is based on a prediction made thereof by Hesiod four hundred years before Aratus, in Hesiod's own poem Works and Days. For more detail on that, see my MoreStories~inUniverse Answer.
3. Absence of the deities?! What're you on about? Without a doubt I saw one recently. And yes, since you bring him up, it was, in fact, Dionysus.
The gods' escape from the world was definitely not the view concerning the behaviour of the deities in the popular culture of the period in question, neither in the Hellenistic period nor in Roman times.
This is so much so the case that a philosopher contemporaneous with both Lucian of Samosata and Pausanias—who argues that Zeus being so unmistakably human-like as depicted in the arts is absurd—seems to be, ironically, the most explicit source of Roman-era sightings of Greek gods, whom he himself claims to have beheld. According to Grant (p. 69):
A later Middle Platonist, Maximus of Tyre (c. 180), is equally
confused. On the one hand, he expresses his belief in divine
omnipotence and proves it by the story that Zeus once tripled the
length of a night. He states that in his own time Asclepius is working
cures, and tells of sailors who heard and saw the god Dionysus,
although other sailors only heard him. He himself has seen the
Dioscuri {Kastor and Polydeukes, mentioned in the Pausanias quote above}, as well as Asclepius, "not in a dream", and Heracles, "a real appearance". He firmly believes in the activity of demons {meaning something more like the Roman genii in this
case}. On the other hand the Dioscuri whom he saw were shining stars
(this makes his testimony difficult to evaluate), and he rejects the
poets' pictures of the gods, which are "credible because of their
charm, but incredible because of their paradoxical nature". Other
stories from the paradoxographical literature are "difficult to
believe", and therefore contain an allegorical meaning.
(With
my emphases and my {notes})
THEREFORE NOT ABSENT
The explanation back then was variegated in a quite similar way to questions like these about deities and their activities now: some take them for granted (moreover reporting experiences therewith), and some are selectively skeptical, while others throw them out almost altogether.
The writings and ruminations of the philosophers tend to occupy mostly the timeframe between Herodotus and the early centuries of the Christian era. For the most part, though, during the period you've asked about—between Homer and Herodotus—gods, heroes and cryptids abounded in the world of mortals.
Dead Heroes Fighting Alongside Gods at the Battle of Marathon
The most vivid instance of the kind of divine appearances we're looking for is actually attested by Herodotus himself, writing about an event which is supposed to have taken place around the time he himself was born. Together with other ancient writers and artists he has a number of divinities featuring as participants in the 490 BC Battle of Marathon.
The Athenian Pheidippides (a.k.a. Philippides), was sent by his fellow citizens to courier a message for help from the Spartans against the oncoming Persian military forces. A professional long-distance runner, Pheidippides completed the journey between Athens and Sparta the day after he had begun it. Upon his return from Sparta to the Athenians he reported to his fellows that, on his way, while rounding Mt Parthenion near Tegea, he had been accosted by the god Pan. The god sent him to ask the Athenians why they did not venerate him, even though he had always been a friend to them and evermore would be so.
The word "panic" is derived from panikos, a seizure of frenzied fear specifically inspired by Pan. At the critical point in the Battle of Marathon, Pan appeared and instilled the Persians with panikos, thus costing them the victory. In gratitude the Athenians dedicated a temple to the god, performing annual sacrifices in his honour, as well as a Lampadephoria, a "torch-bearing" relay race.
Constructed around the same time in the Athenian Agora was a building called the Stoa Poikile, "Painted Porch." Contained therein was a mural portraying the Battle of Marathon, with a depiction of the goddess Athena and the god Herakles fighting alongside the Greeks against the Persians. Also present at the battle was the dead hero Theseus, who had been king of Athens at the time that Herakles was a mortal man. In the painting he is represented as emerging from the Underworld, returning from the dead. Marathon, a grandson of the sea-god Poseidon after whom the plain of Marathon was named before the time of Herakles and Theseus, was likewise present at the battle.
Writing more than 500 years later, Plutarch says that:
In after times... the Athenians were moved to honour Theseus as
a demigod, especially by the fact that many of those who fought at
Marathon against the Medes thought they saw an apparition of Theseus
in arms rushing on in front of them against the Barbarians.
In Pausanias' description of the Stoa Poikile painting, he also mentions a certain mysterious man dressed in rustic attire who used a plough to kill many of the Persians before suddenly disappearing, never to be seen again. Upon searching for him the Athenians were instructed by an oracle to grant divine honours to Ekhetlaios [Echetlaeus], "[He] Of the Plough."
GIANTS
Regarding the giant boxer Kleomedes of Astypalaia, see the section on Superhuman Athletes
in see my MoreStories~inUniverse Answer.
The Makrobioi [Macrobii] and the Syrbotai [Syrbotae]
The modern term Ethiopia comes from Latin Aethiopia, which derives from Greek Aithiopia, which was a very fluid term in ancient times, often referring to the region that is now East African Sudan (rather than the country that is currently named Ethiopia) but sometimes to the entire African continent and even parts of Arabia and India, insomuch as these places were known to the Greeks.
In Homer's time there was believed to be, in Aithiopia, a nation of virtuous people who never died, dwelling in a land of endless summertime and enjoying the company of the Greek gods, who visited them for an annual feast.
Later on Herodotus would write of a group of Aithiopes ["Ethiopians"] called the Makrobioi, "Long-Lifers" or "Long-Lived" Ones, because their average lifespan was 120 years. They, he says, were the tallest, most beautiful people in the world. They were also so wealthy that they fettered their prisoners in chains of gold.
Pliny the Elder, a Roman writing in the 1st Century AD, says that in the vicinity of the Makrobioi lived an Aithiopian tribe called the Syrbotai, whose average height exceeded eight cubits (twelve feet). Krates of Pergamos, writing a couple of centuries previous to this, is Pliny's source for this people. If Herodotus is aware of the Syrbotai, for his statement about the Makrobioi to be consistent, it would have to mean that the Makrobioi are at least as tall as these 3.6m-tall giants, if not more so.
The Kynokephaloi (and the Saint Who Carried Christ)
Going back to the era of Homer, Hesiod writes of the Hemikynoi, "Half-Dogs," a proud race of Anubis-like people who had dogs' heads and dwelt in the same fluidly-defined region of "Aithiopia." Later on they were dubbed Kynokephaloi, "Dog-Heads," which the Romans Latinised into Cynocephali.
As the Roman Empire became increasingly Christianised, these creatures came to be perceived as devourers of human flesh. By this point they were associated with a region somewhere on the border of Libya and Egypt, from which they were called the Marmaritae and were even said to be gigantic monsters.
There are two main variant traditions about the legendary St Khristophoros [Christopher], the "Carrier of Khristos [Christ]." In both, he was a giant who lived during the reign of a Roman Emperor, either Decius (mid 200s AD) or Maximinus II Daia (early 300s AD). In the more culturally Latinised Western Christian tradition, Khristophoros was very anachronistically said to have been a Canaanite giant, one of the Anakim who are mentioned in the Torah.
In the Eastern Orthodox tradition, which is more heavily influenced by Greek culture, Khristophoros belonged to the gigantic variety of the Kynokephaloi. A version of his tale says that he prayed to acquire a completely human form, and his prayer was answered, in which case he became the completely humanoid giant of the Western tradition. Nonetheless many Eastern religious icons depict him, as a full-fledged saint, still sporting his original canine features.
Down into the time of Augustinus Hipponensis [St Augustine], one of Khristophoros' fellow Africans, there was speculation about whether creatures like the Dog-Heads had human souls which could thus receive salvation. In the European imagination for centuries thereafter, this region of Africa was presumed to be inhabited by these Kynokephaloi, whom Hesiod says are the children of the earth-goddess Gaia either by Poseidon or by Zeus's Egyptian son Epaphos [Epaphus].
In Book 4 of his Historiai, Herodotus mentions the Kynokephaloi, locating them in the western portions of Libya together with some ordinary animals, as well as donkeys with horns, and headless people whose faces were in their chests. Pliny calls this headless tribe the Blemmyai [Blemmyae], citing Herodotus' contemporary Ktesias as his source for information on them. The Blemmyai became another favourite of mediaeval European folklore, although the same name was applied also to a nation of ordinary human nomads who occupied eastern Africa before the collapse of the Roman Empire.
IMMORTALS
Regarding the gods Pan and Herakles, and the goddess Athena, see the section Dead Heroes Fighting Alongside Gods at the Battle of Marathon
above.
For more on Herakles, see #3, above, as well as the section Theagenes, Born Divine
in my MoreStories~inUniverse Answer, as well as, in that same Answer, the section Euthymos, the Ghostbuster
for the river-god Kaikinos [Caecinus].
For the god Asklepios [Asclepius] and the twin deities called the Dioskouroi [Dioscuri], see #3, above, again.
Regarding the gods Zeus Ammon and Apollo, see the section Semi-Divine Royalty
also in the MoreStories~inUniverse Answer.
To speak especially technically, Herodotus does in fact mention "Immortals," Athanatoi, in the Historiai 7.31, but alright, to be fair, he is referring to an obviously mortal human military unit of the Achaemenid Empire, particularly under Khshayarsha [Xerxes] I the Great, a contingent whose number was always kept numbered at 10,000 men.
CENTAURS
In The First Fossil Hunters: Paleontology in Greek and Roman Times, Adrienne Mayor2 (pp. 221-222) fleshes out the perspective of philosophers living during and after Herodotus' time:
On Unbelievable Tales was written by a friend and follower of
Aristotle known as Palaephatus (a pen name that roughly translates as
"ancient tales")...
Centaurs ... were a focus of tension between the logic of popular
belief in marvels and philosophers' belief in immutable principles of
nature. Myth accorded Centaurs the status of a viable species.
Empedocles suggested that Centaur-like creatures once existed but died
out as monsters unfit for survival; later, Aristotle and Lucretius
vigorously denied that such hybrids could ever exist...
Plunging into the Centaur debate, Palaephatus ... articulates a
principle of unchanging species: "If there ever were such animals,
then they would exist today." But not only does his statement
contradict ancient knowledge that some real animals had gone extinct,
his wording leaves the door open for relict Centaur "sightings" and
atavistic births that would prove their existence in the past... And
indeed, live Centaurs were reported in the Roman era.
An entire chapter entitled "Centaur Bones" offers more reports of live Centaur sightings. As part of it (pp. 239-240) says:
[H]ybrid beasts combining contradictory categories (including
bird-mammal griffins of chapter 1) were singled out by the circle of
orthodox natural philosophers as impossible. But while Aristotle, his
follower Palaephatus, and Lucretius heaped scorn on the viability of
mixed species, especially Centaurs, writers like Aelian, Phlegon of
Tralles, and others kept an open mind about seemingly incredible
creatures, allowing the interplay of imagination and skepticism to
fill in the blanks of the unknown. Aelian, for example, wondered
whether time and nature might really have produced populations of such
strange creatures, just as the myths claimed. If Centaurs were
actually once prevalent in certain places and not just figments of
folklore, Aelian reasoned (echoing Empedocles) that they must have
been at least a temporary fauna of the deep past.
Palaephatus' authoritative assertion in the fourth century
B.C.—that if Centaurs ever did exist, then they would still be seen
alive—was given literal expression in a rash of Centaur sightings in
the Roman period. During the reign of Claudius (A.D. 41-54) officials
in Arabia declared that a small herd of Centaurs still inhabited
Saune, a remote mountain wilderness infested by poisonous plants.
Despite the danger, one ... was captured and transported to Egypt as a
gift for the emperor. The Egyptians fed the wild Centaur a traditional
diet of raw meat, but it could not tolerate the change in altitude and
perished. The Egyptians had the corpse embalmed and shipped to Rome,
where the emperor Claudius exhibited the marvel in his palace. Pliny
went with friends to see the spectacle: the Centaur was completely
submerged in honey (a common preservative for transporting cadavers
long distances)...
Nearly a century later, through the reign of nine emperors after
Claudius, the embalmed Centaur of Saune could still be viewed, by
special appointment, in the emperor Hadrian's imperial storehouse.
Phlegon... who served on Hadrian's staff (A.D. 117-138), examined the
marvel himself. The Centaur was a bit smaller than what one might
expect from classical Greek art, he observed, but it had a fierce face
and hairy arms and fingers. The human rib cage merged naturally with
equine body and limbs, and its hooves were quite firm.
Mayor herself is convinced that all these preserved remains were artistically contrived hoaxes.
SATYRS & AIGIPANĒS
Fauns are from Roman mythology. Since ancient times they have often been confused with Satyrs, who did not have goats' heads or legs, and were more humanoid (and barely had horns). The Roman fauns were more akin in appearance to the goat-headed, goat-legged gods Pan and Aigipan [Aegipan] (the latter of whom might merely be a form of the former) and the woodland sprites called Panēs [Pans], after the great Pan, and Aigipanēs [Aegipans], similarly after the great Aigipan.
As in the preceding section, Mayor's book relates the issue quite well (pp. 236-237):
Pausanias remarked that a live wild Satyr from Libya was
exhibited in Rome, and Plutarch described the capture of a Satyr in
what is now Albania. In 83 B.C., the Roman commander Sulla was about
to sail from Dyrrhacium to Italy when his soldiers surprised a Satyr
asleep in a sacred meadow, a place where fire flowed from the ground.
The creature looked just like Satyrs in art and drama, and, when
captured and presented to the Roman commander, he uttered a harsh
whinnying bleat.
Satyr sightings lasted into the early Christian era. Saint
Jerome, a contemporary of Saint Augustine, stated that the emperor
Constantine (d. A.D. 337) traveled to Antioch to view the remains of a
Satyr that had been preserved in salt.
At the end of Endnote 8 to this chapter of the book (on p. 326), Mayor says that "Saint Augustine thought he saw a Satyr, according to Cuvier. Rudwick 1997, 233."
Meanwhile Pliny's Natural History has this to say about Mt Atlas in Morocco:
At night, they say, it gleams with fires innumerable
lighted up; it is then the scene of the gambols of the Aegipans and
the Satyr crew, while it re-echoes with the notes of the flute and the
pipe, and the clash of drums and cymbals. All this is what authors of
high character have stated, in addition to the labours which Hercules
[Herakles] and Perseus there experienced. The space which intervenes
before you arrive at this mountain is immense, and the country quite
unknown.
There is nonetheless an encounter with something which is described as both a Satyr and a faun by none other than Jerome, who says that the famous Egyptian hermit St Antony the Great was given roadside directions to an older monk named Paul of Thebes, first by a Centaur and then by a goatish humanoid.
The second creature speaks to Antony and even acknowledges Christ as Lord, all while alluding to the Book of Psalms and the Epistle to the Romans! (And this goat-like beast-man can fly?) As it happens, Antony's response thereto provides what we could call a fourth explanation of the matter in question, or even a blend of all of the current 3.
Antony was amazed, and, thinking over what he had seen [the
Centaur], went on his way. Before long in a small rocky valley shut in
on all sides he saw a little man with a hooked snout, a horned
forehead, and extremities like the feet of a goat. When he saw this,
Antony, like a good soldier, seized the shield of faith and the helmet
of hope: the creature nonetheless began to offer to him the fruit of
the palm-trees to support him on his journey and, as it were, pledges of
peace. Antony, perceiving this, stopped and asked who he was. The
answer he received from him was this:
I am a mortal being and one of those inhabitants of the desert
whom the Gentiles, deluded by various forms of error, worship under the
names of Fauns, Satyrs, and Incubi. I am sent to represent my tribe.
We ask you on our behalf to entreat the favour of your Lord and ours,
who, we have learned, came once to save the world, and "whose sound
has gone forth into all the earth."
As he uttered such words as these, the aged traveller's cheeks
streamed with tears, the marks of his deep feeling, which he shed in
the fullness of his joy. He rejoiced over the Glory of Christ and the
destruction of Satan, and marvelling all the while that he could
understand the Satyr's language, and striking the ground with his
staff, he said:
Woe to you, Alexandria, who instead of God worships monsters!
Woe to you, harlot city, into which have flowed together the demons of
the whole world!
What will you say now? Beasts speak of Christ, and
you, instead of God, worship monsters!
He had not finished speaking when, as if on wings, the wild
creature fled away. Let no one scruple to believe this incident; its
truth is supported by what took place when Constantine was on the
throne, a matter of which the whole world was witness.
- The Life of Paul the Hermit 8
Jerome goes on to describe the Satyr remains in Antioch, mentioned in the Mayor quote above, at the beginning of this section.
OTHER CREATURES
Encounters with mermen (who were called Tritons in Greek myth) and sea nymphs (typically Nereids, who were related to the Tritons via their common ancestor the sea-god Nereus) were as prevalent in Roman times as sightings of Centaurs and Satyrs.
Pliny says that a delegation from Olisipo (Lisbon) arrived in Rome to
inform the emperor Tiberius that a Triton had been spotted in a cave
by the sea. They also stated that a dying Nereid, covered with hair or
fine scales even on the parts that looked human, appeared on the same
shore. The governor of Gaul reported to Emperor Augustus a mass
stranding of Nereids on the Atlantic coast, and Pliny himself heard
from reliable sources that a Triton was sinking ships at night in the
Gulf of Cadiz.
Triton remains were exhibited in Tanagra and Rome.
Mayor 2000: 231-232
Pliny says there was a breed of (really kool-sounding) horned, winged horses in Aethiopia which were called Pegasi. Aaron Atsma conjectures that these were descendants of Pegasus, the most famous winged horse of Greek myth, hence the name of the African variety of these animals.
By the way, during Herodotus' time and long after, gods were still siring offspring upon mortal women. For much more on that, see my MoreStories~inUniverse Answer.
Notes
1. Grant, R.M. 1952. Miracle and Natural Law in Graeco-Roman and Early Christian Thought. Wipf & Stock, Eugene, OR.
2. Mayor, A. 2000. The First Fossil Hunters: Paleontology in Greek and Roman Times. Princeton University Press, Princeton, NJ.