If you look around on the east side of Kvatch you may find a cabin that would be useful. It has a bed, but the storage respawns.
My concept of eastern monks was formed by watching the old Kung Fu TV show. So I'm picturing an itinerant character. I'd say the most appropriate places for such a character to stay would be one of the game's http://www.uesp.net/wiki/Oblivion:Campsites
I play a fair number of houseless characters these days, and, when I do, I often impose a strict rule: they can only heal by sleeping. As a result of this, I've become pretty familiar with these campsites myself, seeing as how they're absolutely essential to heal these characters. But I also enjoy campsites. Most of them are located in very attractive surroundings. I think it's kind of fun to travel around footloose, sleeping underneath the stars.
Most of them are occupied by bandits so you'll have to clear those out before you can move in (but hey, nobody ever said the life of a monk was easy, eh?) The good news is that if you pick one campsite as a base of operations and return to it often the bandits (and any containers in the camp) will not respawn.
Good luck !