To store and organize it how? For carry, or for hanging on a wall, placing on a desk...?
For the majority of ideas that lend themselves towards storing a full sheet of paper flat in some manner -- be it a clipboard, clamshell, desk tower paper tray organizer, wall hanger, or similar -- I think you're going to run into the limitation that it will inherently require at least one flat surface slightly larger than a sheet of paper, be it letter or A4 in your locale. And that is likely to be too big for the printing area on the majority of people's printers.
I know this is heresy on a 3D printing community but there are applications to which a 3D printed object is not practical or economical. An accordion document organizer is only a few bucks at the office supply store, and you can probably even find ordinary folders and clipboards at your local dollar store. If you must DIY it, I think you'd have better luck making some kind of origami contraption out of posterboard or something.