Remember that, as with most things in life, you tend to get what you pay for. Ask about security at the storage building. Ask about fire protection. You'll sleep easier knowing that you've stored your goods at a facility with a sprinkler system and a smoke detection system and alarm. Find out about restrictions on storing hazardous substances.
Some storage units are climate controlled.
This may be important if you have delicate antiques, expensive wine or valuable paintings to store. Your computer may not like big swings in temperature and humidity either. Inquire as to insurance coverage for your goods while in a storage building and be sure you fully understand the terms of the protection you have. If you own items of significant value, it's worthwhile to take photographs of them to have in the event of an insurance claim. Make sure you understand when you can put your possessions into a storage unit and when you can have access to them.