At a couple of hospitals, I've been able to use a call room. But that would be uncommon, and only after you have established a working relationship there. If you have the right temperament, you might see if there is a youth hostel nearby. Those can be a lot of fun and a chance to meet interesting people from all over the world. Perhaps not for a light sleeper on a 12 hour shift though!
Otherwise, try posting a query on Craigslist housing wanted. A roommate who is only there for two nights a week is considered the ideal roommate by a lot of people who otherwise wouldn't share a fabulous home.
I feel it only fair to warn you that if you get audited, you may have to pay taxes on the portion of the housing stipend you spend at home. Very unlikely, but it is certainly possible for the IRS to ask for proof that you did not return home every night if you are audited.