We were in a 1BR villa as well...actually we have rented one several times in our villas. We either stored them in the laundry room (at OKW) or in misc. closets. I think they are thin enough to go under the bed in a regular room though...I'm not positive though. They definitely do not fold. The box will be a little larger than the size of a crib mattress with some bungee cords and lots of plastic wrap.
Good luck, and despite the excess box in your room, I think you are very wise to rent a crib while away from home!