Your fists. That's right, your dukes, your bare knuckles, your twin hammers of justice against the scum of the Wasteland...
In Wasteland, unarmed combat is worked out using two skills: Brawling and Pugilism. Even if you have a level 7 Pugilism skill, you still only get one attack if you only have Brawling 1. Therefore, to be a really good unarmed fighter, you should also take the time to develop your brawling skill a bit. If you like to keep at least one tool of certain types in your party (Crowbar, Shovel, Pick ax) you can use these... although, if you can't scrounge enough Proton axes, you might consider getting some Chainsaws.
The two major advantages of Pugilism are that you don't have to keep a melee weapon in your inventory, and you fight in style: you get to go around beating down your mutant enemies with your bare FISTS. That's about as manly as chugging Brawndo, blasting Van Halen while machine-gunning hordes of Leather Jerks and Blood Cultists from the back of Ace's Jeep.
"Your bare hands. What more do you need?"