If a player wants to do something a little out of the ordinary such as punching down a wall with his bare fists, what should I do? Do I make a DC for it? Do I make it so that no matter how much he rolls, he won't destroy the wall? Should I simply tell him that it won't work?
For things that are physically impossible, don't bother with dice. As DM, dice are for when the group needs to determine whether something happens that's probable but neither impossible nor guaranteed. For impossible things, you're allowed to just say so:
When there isn't a rule allowing something to happen, it's the DM's prerogative to judge whether something is possible, and how difficult it is. Things that are very easy like "I walk up the stairs" don't need rolls for you to say "OK" to — neither do impossible things need rolls for you to say "No" to.
Nota bene, for truly "out of the ordinary" but not impossible things there is Difficulty Class and Damage by Level (DMG, p. 42), which gives you a framework for assigning DCs to tasks that aren't otherwise covered by another rule in the game. Punching down a wall with bare fists shouldn't ever need a DC, but punching down a wall while accidentally turned into a golem might call for a high DC, and that's where to find it.
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It depends upon the impossibility level, and the nature of your campaign.
For example, punching one's way through the stone wall...
If the campaign is realistic or semi-realistic, just warn them it's impossible, and if they try anyway, instant damage from their punch.
If the campaign is kinda epic, assign an absurdly high DC (like 50+), but allow the optional "reroll at +10 more on a nat 20" and let them abort before rolling.
If your campaign is in silly mode, Just let them succeed on a Nat 20, but the wall slaps them on a nat 1.
In cases where the character is ill equipped, but the action is theoretically plausible, let the player know what they need to succeed. For example, Leaping a mountain. Even in most silly mode games, it's not going to happen. Figure out what they can achieve, and on a really hot crit, let them go a little bit more, but don't violate the cartoon physics any more than you need to.
And, despite the recent raft of GMing advice to the contrary, it is okay to say "No, sorry, doesn't work that way."
Tell them you think it's impossible; they might have a clever approach that they'll tell you about, or you might disagree about the power of their character. If they persist in their desire, ask them why they want to do it.
If someone wants to do something like punch down a wall or parlay with the clearly non-intelligent ooze or take a well into the dungeon with them, it can be a sign of a larger desire. Maybe they want to try a different character and are trying to kill them off, in which case you can give their character a more noble death. Maybe they're desperately trying to make the game interesting, in which case you can give them more things that appeal to their playstyle.
One thing you shouldn't do is punish them for their attempt. If they're trying to eat glass, sure, they'll take damage. If they punch the wall, the first few times they're going to get hurt knuckles but not shattered bones. No need to assign damage unless that's their goal or the action is dangerous anyway.
In my group we have a rule and it's 'always let them make the roll'. It's simple. They try to knock down the wall. Set a DC, becuase that's how the universe works. Just give them a DC they can't make. No matter how high they roll you know they can't succeed becuase given the rules of the universe they don't have ability to do it. After the player rolls a few times and you tell them they're not making any progress, they know it's impossible.
While it sounds railroady, and kind of is, what it actually does is set that the universe you are playing in exists within a set of rules and some of those rules (In the case of the wall, those of basic physics) are not mobile.
That's just the way I run things.