Since we can't put storage items like bags inside other storage items, how about a new special storage chest? Basically it's a storage item that's only for keeping other storage items, but with a behaviour that's akin to the cabinets in houses: you'll be able to store items in it, but you won't be able to use those items while they're stored inside.
This would be very helpful for things like long quests where you have to lug around a bunch of stuff that you keep in separate bags but really don't need quite often, or when you decide you're ready to move to a new house and have a lot of stuff to move.
And since storage items inside the chest aren't really active while they're inside, there's no significant Backpack UI change needed to implement it—only some internal flags for the storage items themselves and a new storage-only guard for the chest's drag-and-drop and accumulate filters.