I am a fan of this song, and this question. After thinking about the song for some time, I was enabled to gain some impressions of my own, and I felt inclined to share them here.
The song seems to be about a child's becoming or wanting to become a superhero. To me it says that the author of the lyrics found a loving protector, the female character in the song, before he found stardom. I am not sure if "she" is his mother, or Jesus, or a special woman in his life, or an alien entity. But I think the song expresses a child's feelings of unworthiness about receiving the promised gifts, and it assuages his concerns about the risks he will confront on the path that he is being shown.
The lyrics seem intentionally childish. I think this can be heard in the opening line, "I've been reading books of old" - in the contrast between the casual "I've been reading" and the archaic "books of old". Chris Martin is not, as others have pointed out, trying to prove himself as a classicist here - he studied under Jeremy Bentham. If it sounds like an imaginative child is putting his own spin on the myths, then we should at least credit the musician's mythical associations for setting a tone that is consistent with the rest of the song. There is even an animated official lyric video which shows a child in a spaceship.
At the same time, it seems possible to synthesize the other answers into an interpretation that refers to something more mature and more specific, and something which may even be deliberate on the part of the artist.
Consider that Achilles was known for being swift, and Hercules for being strong. The narrator is worried that he won't be swift or strong enough to make his dreams come true. But his benefactor replies that she's not looking for someone who is already a hero. Maybe she then reassured the narrator with the verse from the Bible that says: "the race is not to the swift, nor the battle to the strong". The full text of that verse is:
Ecclesiastes 9:11 I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.
The verse lists five superhero gifts and traits:
- race / swift
- battle / strong
- bread / wise
- riches / understanding
- favour / skill
The song lists five superheroes, and things they have:
- Achilles / gold
- Hercules / gifts
- Spider-Man / control
- Batman / fists
- Superman / suit
As I already noted, there is a good correspondence between the first two items in each list. For example, although it would sound just as well to switch Achilles and Hercules in the song, this would break the correspondence to Ecclesiastes: Hercules is known for being strong, not swift; it is Achilles who is called "swift-footed".
Now, of the modern superheroes, Batman (number 4) is best known for being wealthy ("riches") in his day-job identity, and he is the next easiest to cross off the list.
As for the last item, number 5: Superman may have just as much "favour" as the other heroes, but this depends on his suit, which serves to insulate his superhero identity from that of the timid journalist Clark Kent. You wouldn't want to try to rescue someone while dressed like Clark Kent! It could have been Spider-Man's suit, or Batman's, but we can see that wearing a suit brings (worldly) favor, and making a suit requires skill. Certainly a suit is more apt to win favor than the "race" or the "battle".
Incidentally, creating a metaphorical "superhero suit" might be an essential skill for a pop star, who must also maintain two separate identities - the reclusive intellectual composer and lyricist, versus the "superhero" stage performer. Some of the symbolism that we're trying to analyze here might be like a costume that creates the lyricist's public persona: something that covers the details of his private life, without pretending there isn't a human beneath it.
The hardest connection for me to explain is Spider-Man with bread and wisdom. The best explanation I could think of, is that Spider-Man's abilities are due to an "unstable performance-enhancing chemical"; in fact he is the only hero in the song's list whose abilities are said to come from an ingested (injected?) substance; and bread is the only edible item on the Ecclesiastes list. Moreover, bread is perishable and therefore "unstable". Finally, in the Bible there is a description of a mysterious unstable substance that Moses called "bread" but his followers called "manna" or "What is it?":
Exodus 16:15 And when the children of Israel saw it, they said one to another, It is manna: for they [knew] not what it was. And Moses said unto them, This is the bread which the Lord hath given you to eat.
We know that manna was even more unstable than bread, because it grew worms if left "until the morning" (Ex 16:20). I'm not sure what this has to do with "Spider-Man's control", except that after Moses gave manna to his followers and taught them how to gather it, they were thirsty and hungry and murmured against him - and they still had to eat it for forty years. Did they all become superheroes before they got to the promised land? Anyway, Moses is the best connection I could make between bread and Spider-Man. Also, control may be connected with wisdom, the corresponding attribute in Ecclesiastes.
Besides the five gifts, there are a few other hints at Ecclesiastes 9:11 in the song. In fact the entire rest of this verse appears in one form or another in the song's lyrics. Ecclesiastes 9:11 mentions "time and chance": "time" becomes "somebody I can miss"; and "chance" becomes "risk". The phrase "I returned, and saw under the sun" becomes "The moon and its eclipse". The song's words "moon and its eclipse" originally suggested a lunar eclipse to me, but with the help of the Ecclesiastes correspondence, I can now see that the song is talking about a solar eclipse.
In answer to the original question, I think the song is not so much a reference to specific super-heroes and their stories, but to five super-hero traits that the artist was seeking to possess as a child. For whatever reason, the song's list of five traits seems to mirror the list of five traits in Ecclesiastes 9:11.