Storage is similar to your Mog Safe except that it can only be accessed from Mog Houses and Rent-a-Rooms, not Nomad Moogles. Storage is increased by placing Furniture in your Mog House Layout. Every piece of Furniture has at least one storage, but some have more, allowing you to maximize your storage space without taking up as much room in your Mog Safe.
While you can place as much furniture in your Layout as your Mog House allows, your Mog House can have a maximum of 80 storage spaces. Any additional storage is "wasted". Additionally, while you can place any type of Item in your Mog Safe or Mog Locker, your Storage has one restriction: you cannot store Furniture items in your Storage. (Why are you trying to hide that bed under that flowerpot, kupo?)