The storage room allows a vault to hold more junk, outfits and weapons that are not currently in use. Vaults start with a base storage capacity of 10. The storage room requires 12 dwellers to be unlocked.
Despite being marked as an Endurance room, having an Endurance-oriented dweller assigned to this room appears to have no effect on the room's function. It does make that dweller happy, though that dweller will not gain levels while assigned to the storage room. You might want to use storage rooms to stop your newly grown-up dwellers from wandering around your vault and have a place to find them. Empty rooms are vulnerable to radroaches; having a dweller occupying that room allows for an infestation to be controlled to an extent.