Cockroaches have easy access through cracks in your storage unit. The cracks could be in the foundation, the walls or between the door and the floor. The bugs also like to infest cardboard boxes. A box you obtain from a local store or a neighbor could already have roaches living inside it. They also hitch rides on grocery bags.
Storage units provide an ideal location for cockroaches to dwell. People are not normally in the units, which translates to a lack of light --- a boon for cockroaches, which are mainly nocturnal. Storage units with heat or in warmer climates will also provide an ideal location, as cockroaches do not like cold temperatures. Once in the storage unit, they will search out and dwell in the cracks in the walls and the floors.
Cockroaches love to eat starches, sugars and grease, but you probably do not have those items in your storage unit. Cockroaches can dine on other items that are not food for people. Books will attract the bugs, and they will feast on the starch in the bindings. The old leather chair you want to restore will also give them something to dine on, as roaches will consume leather. They will also eat glue.
Keeping your storage unit clean will help greatly in your battle against cockroaches. When you encounter a cockroach in your storage unit, purchase a roach killing spray and use it as directed on your unit. The spray will offer immediate results. The roaches will come back, so also buy roach-killing baits, which last longer. Another roach control product you can buy is boric acid, which is not very toxic to humans and keeps killing for months.