Not super cheap nor super fancy, but such is the market for covered storage in Anchorage. Have used these guys a couple of times, and been universally pleased with their customer service... Every staff member has acted like they want to be helpful, which might seem like a low bar, but is enough to stand out from the crowd to me. And they'll refund a partial months rent if you vacate mid-month, which seems like the sort of common courtesy which ought be universal, but again, in my experience has not been.

Easy to get a storage unit, set up automatic payments, get a different unit, and to check out from the unit. The staff were very friendly and helpful. We had a unit here for about 1.5 years and everything went smoothly.