With any moving storage arrangement, make sure you get a written agreement. Most companies will be happy to provide one, and most prefer to do so. Read the agreement and get any questions answered to your satisfaction before you sign on the dotted line.
If you've hired a moving truck to move some of your belongings to storage, be sure to ask the company what size truck they'll be sending. This is essential, especially in smaller, cramped facilities where there might not be enough room for a big truck to maneuver.