Personally, I lift in my garage most of the time (go into local Crossfit gyms to shake it up every so often).
However, I have lifted in a storage unit. Granted, we had stuff in there, but it definitely is a viable option. A couple things to keep in mind, though: no lights in the units, usually; and I would definitely let the management know about it. Most of them want the units rented, so usually won't mind too much, provided you don't damage the unit itself.