I can't give a well-sourced answer as to how much skill it requires to shoot through 12 axe heads, having never done more than basic archery myself.

However, if I remember correctly, the text implies that the suitors wouldn't even make it that far in the task, as the true test would be stringing Odysseus' bow, which required a crazy amount of strength.

So...if you don't mind me adding a second question: assuming that the maker is very strong, with the materials and tools of the time, would it have been possible to create a bow that has enough resistance to be "unstring-able" by a normal person, but flexible enough material that it wouldn't snap in half when Odysseus did it? (Might be more of a physics question, sorry. XD)

Either "something an experienced archer could do" or "superhuman and physically impossible", depending on how you read it. No one knows what the contest actually involved. You say that the arrow has to go " through the holes of twelve axe heads", but that's already a translator's interpretation of the contest. Here's what the text actually says: 19.578:

διοϊστεύσῃ πελέκεων δυοκαίδεκα πάντων

...whoever shoots through all twelve axes

At 21.114 and 21.126-7 we have people trying to string the bow and διοϊστεύσω/-σειν τε σιδήρου, "shoot through the iron"; at 21.328 the phrasing is διὰ δ' ἧκε σιδήρου, also "shoot through the iron".

The description of Telemachos setting up the axes is at 21.120-2:

πρῶτον μὲν πελέκεας στῆσεν, διὰ τάφρον ὀρύξας
πᾶσι μίαν μακρήν, καὶ ἐπὶ στάθμην ἴθυνεν,
ἀμφὶ δὲ γαῖαν ἔναξε.

First he placed the axes, after digging a continuous trench,
a single long one for all of them, and made (them) straight in a line,
and he pressed the earth around them.

And Odysseus finally shoots through the axes at 21.419-23:

τόν ῥ' ἐπὶ πήχει ἑλὼν ἕλκεν νευρὴν γλυφίδας τε,
αὐτόθεν ἐκ δίφροιο καθήμενος, ἧκε δ' ὀϊστὸν
ἄντα τιτυσκόμενος, πελέκεων δ' οὐκ ἤμβροτε πάντων
πρώτης στειλειῆς, διὰ δ' ἀμπερὲς ἦλθε θύραζε
ἰὸς χαλκοβαρής.

He took the string and the arrow-notches and drew the bridge (of the bow),
on the spot, sitting in his chair, and shot the arrow
aiming in front of him, and did not miss any of the axes
from the first στειλειή, but it went through to the doors,
the bronze-heavy arrow.

The interpretations of what's going on that are taken at all seriously are:

1. Shooting through gaps of some kind in an intricately-shaped axehead, or a gap between a curved axehead and the shaft; this could be suitable for mid-Bronze Age Minoan double axes, but we've got no evidence of anything similar in the Iron Age.

2. Shooting through metal hanging-rings (ὀγκιον) at the butt-end of the axes; however, there is no mention of ὀγκια anywhere in the text. The support for this interpretation is that στειλειή (untranslated above) may mean "shaft" or "haft" of an axe; but the word's meaning is uncertain.

3. The axeheads are detached from the shafts and the task is to shoot through the sockets; the openings would then be tiny, and it'd be basically impossible to verify that the arrow actually went through any of them without a high-speed camera.

4. The idea is to shoot an arrow so powerful that it breaks through a metal axehead. This has the merits of (a) taking the repeated phrase διὰ σιδήρου ("through the iron") literally, and (b) it becomes more a feat of strength than of unverifiable aim, which is fitting with the previous task of stringing the bow (something that requires strength, and which only Odysseus -- and Telemachos -- can do). It is also potentially linkable to Egyptian pictorial representations of pharaohs shooting arrows through sheets of metal.

It's a superhuman task if the openings are too small, if the axes placed too far apart, or too close to the ground, or if we go with option 4 at all. That's not an obstacle: the idea of the contest, the setting, the context, and the genre all encourage the idea that Odysseus is performing a superhuman feat beyond the reach of any living mortal.

Having said that, a successful effort to replicate the feat has been reported with openings 4 inches (10 cm) in diameter and placed no more than 1 foot (30 cm) apart, using a bow with a 47 lb draw (which, depending on various factors such as draw length and arrow weight, might perhaps equate to an arrow speed of ca. 50 m/s or 164 fps). Smaller openings, or more space between the axes, create physical problems not of whether it's possible to aim that accurately, but of the arrow being pulled downward by gravity. It has to be said, 10 cm is a pretty wide opening.

(For reference, a distance of 3.3 m from the first axehead to the last, with a 50-cm-long arrow going 50 m/s, means a drop of about 3 cm from the moment the arrowhead reaches the first axehead to the moment that the tail passes the last axehead; add on to that problems of airflow and turbulence, and you've got a pretty challenging task.)

But I reiterate that the setting needs to be taken in the context of its mythical setting, with a hero who is greater than other mere mortals achieving a feat that is supposed to be outrageous. For that reason I find option 4, above, quite attractive. It's possible to imagine an arrow fired so hard that it breaks an iron axehead; someone like Odysseus, then, would be able to get an arrow through twelve of them! It's also a story with strong overtones of tradition: there are also bow contests for the hand of a bride in the Indic epics the Mahābhārata and the Rāmāyana, viz. Arjuna competing for the hand of Draupadi, and Rāma for the hand of Sita, and in both cases the point is that Arjuna and Rāma show that they are able to do something that no one else can -- in Arjuna's case, by stringing a bow that no one else can string and then shooting five arrows through a "wheel" at a target; in Rāma's case by drawing on a bow that no one else can draw, and drawing so hard that it shatters.

Your question is still interesting but I think it should be understood that the real test is about 'who can use Odysseus' bow?' rather than one's marksmanship skills. That's why the suitors desperately attempt to string the thing, but they're too weak.

This is not an historical question as it is about myth, but I'll give it my best shot. I have taught the Odyssey for almost a decade, and as far as I know no one really knows what that contest is. Does shooting through axe heads mean shooting through the metal? Or are they all aligned so that the arrow passes through the eyes? In that case, what is the significance of 12 of them? Being able to shoot an arrow straight? Are they single or double headed axes? I want to note that the translator /u/xaxers refers to decided the arrow went through the eyes. Other translators disagree (I'm looking at Fagles now).

I have a little Freudian theory I try out on my students. Sometimes they buy it. As I said, I don't think /r/askhistorians is the place for this question, but I submit my theory for your judgment.

In book 2, Telemachus summons an assembly, for the first time in 20 years (since Odysseus left). His goal is to secure a ship and supplies to look for Odysseus. We learn society in Ithaca is a wreck - the suitors are not respecting the customs of courtship, Penelope has not returned to her father's house as she ought, Telemachus is unable to demonstrate manhood. All this because Odysseus is gone. In the assembly Telemachus describes Odysseus as a father to the people, hence Penelope would be the mother.

In book 21 with the contest of the bow, Penelope proposes to marry whoever can string the bow and complete the challenge (whatever that is). Telemachus demands to participate. Telemachus, Penelope's actual son, is competing with the suitors, Penelope's figurative sons (as described in the assembly in book 2). Telemachus can actually complete the challenge, but recognizing Odysseus as his father, accepts his authority as father and declines to contest possession of his mother. The suitors, figurative sons, continue to contest possession of the mother because they don't recognize the father, and they all end up penetrated. In book 24 Odysseus and his family confront the villagers, society is restored, and everyone doesn't quite live happily ever after.

So the contest of the bow, in my reading, is not about military skill or physical strength, it's about who can recognize the proper authority (the father!), and who gets to possess women (the father!). Telemachus can make the proper recognition and so bows out and survives. The suitors cannot. Killing those that cannot restores civilization.

I want to point out the Freudian imagery of lubing up the bow in advance of stringing it to, you know, project something.

I'm not a Homeric scholar but I think this is my own crazy idea. If anyone has seen it before I'd appreciate references.

EDIT: Also, this act didn't prove Odysseus' identity. That was the bed trick.