haha! Found this thread without even noticed the tag 😂
Depends on how often you want to access them/how concerned you are about the look.
My immediate thought was magazine files. Box files if you want the plastic files completely hidden but I probably wouldn't have put the papers in the plastic wallets if I was using those.
I've got a couple of the really useful box with suspension files - those are in a storage room and only used for archiving, again though if you've already sorted into the plastic wallets...
I've got low profile magazine files from B&M at work that I use for keeping documents in those type wallets. They work great if you really don't mind seeing the plastic wallets and it keeps them very tidy and very easily accessible. Problem is you can only get what is on the shelf in B&M when you visit...took a few visits to get what I wanted. I've got four of the blacks ones on top of my piano to store music too - the magazine file itself virtually disappears.
https://www.bmstores.co.uk/products/colour-magazine-file-289981
I had these in a teenagers bedroom to store revision guides and all the other stuff they wanted on their desk - which was quite a lot...white because that went with the decor
https://www.ikea.com/gb/en/p/kvissle-magazine-file-set-of-2-white-60203957/
For the kind of things you're talking about storing I have those in either a cupboard with the cheap build yourself basic brown cardboard magazine files (other paper work is in those metal office drawer units).
Anyway, as you can tell I've heavily biased towards magazine files for what you're talking about.