The two games that immediately come to mind are Fallout 3 and No More Heroes, but for different reasons than I've seen here.
Every once in a while in Fallout 3, I would be crossing the wasteland, usually in the western side of the map, and I would notice that I hadn't seen any enemies for a few minutes. I would stop and look around, and it would suddenly dawn on me just how alone I was. As far as gameplay, that isn't great, but I think it's the closest example of life in a nuclear wasteland I've ever played, if only from the feeling of isolation. The other Fallout games are more entertaining, but if something like the Great War ever happened in the real world, survivors could live the rest of their lives without seeing anyone else. Metroid Prime felt the same, with every living thing you find trying to kill you. I get the same impression from Journey, but I haven't played it yet.
In No More Heroes, every assassin left an impact in one way or another, but the one that affected me the most was Dr. Peace. You meet him to fight in an empty baseball stadium, and when you walk in, he's standing on the pitchers mound singing into a microphone. When he finishes, without even looking at you, he starts discussing a meal he had the night before with his wife and daughter. You just get the feeling that he knows he is about to die, and his last wishes were to say goodby to his family, and sing at a baseball game. It became one of those cases where you really don't want an enemy in a game to die, but it's the only way to progress. It's a similar case to the Boss, if not as dramatic.