There seem to be two distinct topics going on here: Damage to weapons, and damage to armor, and the subsequent failure of both. Here's my $0.02 on both.
Weapon damage:
This should really only come from direct damage to a weapon. A raider with an axe could smash your rifle's receiver, a sharpshooter could plink a gun out of your hands, but using a weapon shouldn't incur gradual failure. The only thing I'd perhaps do is have guns get dirty from use, which they do. All you'd have to do is clean it, which would require, say, a modest repair skill, or a high skill in the weapon category. (Just for the sake of argument, cleaning your pistol would require, say, 45 repair, or 85 small guns, assuming those two skills are in the new skill set. A tool kit would add only 10 to your bonus because generic tools aren't really made for gun maintenance, but a gun cleaning kit would give you 25 bonus since it's made for that and presumably has clear directions.) A fouled pistol would be less accurate, and the chance of it blowing up accidentally should go up a little, but otherwise this shouldn't be a major gameplay element. I'd keep the very infrequent critical misses where you blow up your gun or loose your ammo, though.
On the subject of Armor taking damage, we've got a bigger problem. Some types of armor lend themselves to maintenance. Metal armor made after the wars is basically plates of metal bent and shaped into armor, perhaps even without aid of a forge. Repairing it or even creating it shouldn't be too difficult with access to a scrap yard or similar. Body armor like a policeman wears is kevlar with plates at key areas, it takes a LOT of damage from a bullet, and some bullet can just go right through it. If you found a police armor vest, it wouldn't last you too long in a firefight.
Armor could be broken down into types for different parts of your body. I remember in FO and FO2 the screen would still say you were hit "with no armor there" even when you were in Power Armor and you had armor everywhere. The Vest I mentioned would only cover your torso but protect you very well from bullets, but it would need its trauma plates replaced every so often, once they soaked up too many bullets. This would have to be kept track of, and thus it would make using such a vest a pain for the player.
Other types of armor would be different; you're not going to do anything shooting Hollow-point rounds at Power Armor or even metal armor, they should really just go "splat". Then we bring up the thorny issue of what happens when someone just happens to have an anti-tank rifle than can put neat little holes in your Power armor? You certainly can't repair it easily; it was designed at a time when it would have had a support structure of people who would've taken care of it, but now it's out on its own with its user. Given the scarcity of Power Armor in the Fallout series, if armor damage was universally implemented, Power Armor would end up looking like the surface of the moon.
Here's a possibly solution: Make common armor like leather jackets, leather armor, and body armor able to be damaged by certain things and levels of damage (For instance a leather jacket won't suffer much if some's going after you with a bullwhip or a lead pipe, but it might get cut up if someone's after you with a knife. Likewise, thick leather armor will fend off the knife, but it too could get ruined by someone with a finely honed Japanese sword. Flak vests would take damage to their trauma plates, which would require replacements. If the composite plates weren't available, you could use whatever metal was at hand, decreasing the effectiveness as appropriate. More expensive armors like Combat Armor and Power Armor could be very, very difficult to destroy. Normal bullets and blows just wouldn't cut it. Not even standard AP rounds from a normal gun should be able to cut through power armor (since the idea is to make the wearer immune to normal guns). Make a few select weapons able to damage Power Armor; perhaps an anit-tank rifle or an anti-material rifle like a Barret light fifty, but nothing else, unless the difficult maintenance of Power Armor intrude on the gameplay and enjoyment. A few places in the game where the player has to be cautious or creative because his foe just happens to be packing the right weapon.
Anti Power Armor weapons wouldn't be everyday occurences; during World War II a whole variety of weapons appeared to take out tanks; paging through "The Encyclopedia of Weapons of World War Two" reveals precious few weapons of much use agains Power Armor; the only rifle able to penetrate over an inch of case-hardened steel was the cumbersome, heavy Japanese Type 97; this weighed 114 pounds and had a 7 round box magazine. The odds of something so unique existing in the wastes are low.