Yes, at least where I use one. We own a small retail business and found this year that we needed seasonal storage, and so rented a unit this fall. The place has excellent video security and keeps a record of every entry and exit, both by code and camera. I don't know how long they keep the video records, but given the ease of digital record-keeping I'd bet the key code transactions are held indefinitely. They also make it next to impossible for more than one vehicle to enter the place at a time. You could, however, enter on foot without using the key pad, when the place is open (it closes at night), but you'd still be recorded visually, and probably stopped by an employee.
Other storage places are open 24/7, but from what I've seen locally, they pretty much offer similar kinds of security arrangements.
caw