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.
For the SPECIFIC example of breaking down a wall with your bare hands, this is handled directly with RAW: a 1ft-thick stone wall is a DC 35 Strength check, and a 6in wooden wall is a DC 25 Strength check. (DMG p35, RC p175)
At level 30, a character who started the game with 8 Strength and never increased the ability (except at each tier increase, where every ability goes up by 1) would have +15 to a Strength check, and would be able to meet the DC 35 for a stone wall with a natural 20.
A character who begins the game with 20 Strength (allocate points for 18, +2 racial mod) can meet the DC 35 at level 16 if Strength is being increased at each opportunity. (level 1: 20, level 4: 21, level 8: 22, level 11: 23, level 14: 24 – 7 Strength mod, plus 8 for 1/2 level at level 16)
Some items can enhance Strength checks:
Some rituals can enhance Strength checks:
Some powers can enhance Strength checks:
Additionally, an Eternal Defender (Fighter Epic Destiny) can get a +10 untyped bonus to a Strength check once per day.
Here's a build that can guarantee hitting the DC 35 at level 14:
Korgnath has +26 Nature, and Ritual Sacrifice can give him an additional +5 until his next extended rest, guaranteeing that he gets at least a 30 on his Nature check when performing the Feat of Strength ritual.
The Steady Strength ritual permits Korgnath to take a 10 on his Strength check.
If Korgnath rolled 1-8 on the Feat of Strength ritual, he has +27 to his Strength check to break an object once per encounter, and when he takes a 10 the result is 37, which is enough to break the stone wall. If Korgnath rolled 9+ on the ritual, he doesn't need to use the encounter power of his gauntlets to reach +30, so he can continue to smash walls bare-fisted for the full encounter. (Note that casting both rituals requires a total of 11 minutes' prep time.)
If Korgnath only needs to smash wooden walls (DC 25), he only needs Steady Strength (1 minute prep) to guarantee destruction by taking a 10. (Of course, he could always just roll 2+ without using the ritual.)
In general I avoid saying something is impossible.
Want to punch down that wall? Sure, go ahead. Start rolling damage. The wall is a lot harder than your fist, though, I'll be generous and only apply the same damage to your fist as to the wall.
Since I don't know the 4e rules I don't know exactly what's happened with hardness but I would figure the wall has some, your fist has none. Unless you can do some really nasty damage with your fists all that's going to happen is you beat yourself up, nothing happens to the wall. (Note that if the wall is made of something flimsy success is quite possible--a fighter type certainly could punch through modern drywall.)
As for the fly to the moon example in the thread--assuming they have some means of flight at all I'll let them try. When they start making constitution checks a few miles up they'll probably change their tune. Also, if they're using something wingish to fly I'll start requiring strength checks as the air thins out (less to flap against.) If they're using something that doesn't flap and have something that negates the need to breathe they'll eventually find out just how vicious sunlight is without an atmosphere to filter it. They'll also find out that you can't drink without an atmosphere.
If they have something that doesn't flap, no need to breathe, no need to drink (solid enough food will still be able to be consumed, albeit very unpalatable. I'm not aware of anything that negates drinking without also negating eating anyway) and radiation resistance (there's nothing in the game but I would permit them to research it) they'll find out about sleep deprivation as if you stop flying you'll fall. (A flyer will NOT be in orbit.)
Overcome that (say, a ship where they can take turns flying it) and a year later they'll actually get to the moon.
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.
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.
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."