A little while ago, I decided to beat Fallout 3 only using the Unarmed skill in combat. I managed to do that almost exclusively, although there were a couple times I used grenades on people on high unreachable ledges.
(At the end of the game, I did a little posing with the Fat Man, a portable nuke launcher, one of the more endearing items in the game.)
One thing I noticed during the run is that I tended to get hurt only on the optional sidequests. After that, I largely stuck to the central story. I managed to win the game at only level 11. If I do another challenge run, I'm going to see how low level you can be at the end of the game. You can skip a great deal of even the central quests, and you can get an ally to fight for you in the first major city of the game.
I'm not sure, but I think the only real source of XP would be the rewards for completing quests critical to progressing the story. It's impossible to avoid that, so that'd be the minimum XP progress. Since I would have an ally, the already easy battles should be no trouble at all.
Now playing: The Cast Of Sweet Cuppin Cakes - Sweet Cuppin' Cakes Theme Song