Besides clogging your locker with things, there are plenty of areas you can store stuff in. If your museum rooms are available or empty, you could always put extra sets or misc things in there. If your mail storage is relatively empty too, you can attach items to the letters and put them in there (although it's a pain when trying to find specific things). Besides those, using items as decor outside in your town might be an option, depending on the item and your town theme of course. And there's always the beach and private island, assuming your town has one, which are useful if you block it off from guests.
^ If you use those tips and still run out of space, another idea is to sell or discard all orderables that you aren't using, since you can easily buy those back through the catalog, except for crowns and thrones of course.
And sometimes, storage alts are the only way to go. You can always use their houses for storing extra furniture sets, and each extra character has mail/locker storage too so it is always an option you should keep open.