We use cane baskets here. The main toy basket doesn't have a lid, but it still looks all neat and means I can just throw things in there in a hurry, and he can pull them out without needing my help. Our other cane/wicker/seagrass baskets have lids.

If your kids aren't using all that stuff at once, could you be bothered keeping more stuff in their rooms and rotating?

We also have an IKEA Expedit, it's our bookshelf (with a few cane baskets in there hiding magazines and the iron. ) - but a small one of those could be the go too - since it could be used as a workspace for drawing, trains, cars, and then things packed away into the cane baskets at the end - it wouldn't be your cheapest option, but it will always be there as storage/bookcase you can use.
I've considered that kind of thing as a clothes sorter even!