Paper has its advantage in that it still breathes. However with thoroughly dried seeds and stored cool (fridge/freezer) glass jars and plastic shouldn't be a problem.
I'm thinking of - when my potting shed is finally built (sometime...) - a sort of spice rack, storing the seeds inside tiny glass spice jars (like ones from Ikea) and hanging them from a top bar in a low fridge. The bar makes it easy to slide the different racks aside to acces others.
Store according to flower/veg, genus/species or just alphabetical....
Ofcourse this is all to your choosing. At the moment I'm storing the few seeds I have in glass weck jars in a cool and dark unused room and they're doing fine.
Most important is that it's flexible and to your liking