How can I use the backs of doors for storage without making the room look messy?

Try an Elfa over the door organizer with mesh baskets — you can configure as needed and the mesh bins hide things well.

Choose storage that earns its footprint: it should make the room easier to use, not just add another piece of furniture. Keep visible surfaces restrained and use matching finishes where organizers remain in view.

My final check is practical: can you find what you need quickly, put it back just as easily, and see when the space is getting full? If not, simplify the categories or move the storage closer to where you use it. An organizing system should make life easier, not become another chore.