I make far more prints than I mount or sell, so I tend to buy only as much material as I need for some specific purpose and have a minimal 'stock'. This is only practical as I'm lucky enough to have a well stocked art-supply shop within cycling distance. I carry the boards back home (yes, on my bike, and when it isn't raining or very windy) in my very large portfolio-case. Unused boards live in the case, except when I'm using it to transport mounted work obviously.

Offcuts and "other" bits and pieces are wrapped in a clean dust-sheet, inside a huge flat cardboard box which once held a table-top. This lies horizontally, coincidentally also under the bed, on top of storage boxes. Under the bed might be the largest undisturbed area in small apartments ?!

Keeping the board completely flat seems to be achievable only when it is stored horizontally and leaning a mounting board against walls quickly gives a slight curl unfortunately. You could look at (clean) painters dust-sheets for protection of whatever large box you can find, but my solution is pretty much the same as yours I think.