How much space do you need? It depends on what type of wines you collect and how you store them. Generally wine storage is offered by the cubic foot or case. A typical case of California cabernet is one cubic foot. Of course pinots tend to require slightly more space. And if you have wine in wooden crates or large format bottles these will require more room.
Many people will use racks two deep for longer term cellaring of wine. You can purchase your own racks or some facilities may offer an inexpensive wine storage system for rent. A good rule is to assume 7-8 bottles per cubic foot of storage or .6 cases x cubic feet of locker. Take into consideration your expectations for increasing your collection. Ideally a facility should have locker sizes that can accommodate a growing or shrinking collection, but you probably will want to avoid needing to move lockers for a few years.