Vatican City has no tourist accommodations. The surrounding area of Rome has a few. Some places are really nice, but our budget didn't allow us to stay there.
We stayed at the Pensione Ottaviano about 10 blocks from Vatican City (Linea B:Ottaviano). For $15 a night in a dorm setting it was fine. No meals are served and they provide a lockbox for anything valuable. Although, why you wouldn't keep all your valuables in your moneybelt is beyond me.
Overall a nice place with travelers from all over so you will have a chance to experience some other cultures while being absorbed by Italian culture.
Another great hostel in Rome is the Beehive. It isn't near Vatican City, however. You can check out their accommodations at www.the-beehive.com.