Today, I routinely download and save PDF/OFX/QFX for every financial and insurance statement, as they become available. The few important paper documents I receive, go into a "to-scan" pile, which I try to clear out every few months.
(I really need to set up a scanning appliance/station that takes only seconds to use. I only need a to-scan pile because it currently takes about 10 minutes to set up for scanning and put things away after.)
Since Apple started providing better end to end encryption for Notes and iCloud, my wife and I just use that. We also have a ton of storage on OneDrive, so periodically I ZIP up important stuff, encrypt it and put on OneDrive.
Fujitsu ScanSnap is a highly recommended product. I have one on my desk that I bought over ten years ago -- it's slightly larger than a box of facial tissues (kleenex), all I have to do to scan is open the two "flaps", insert document, press a button, then assign a file name and folder (easy once a history list is established). Scans color or B&W, single side or duplex.
For digital stuff it’s mostly the same idea. If it’s email (which is almost always the case), I archive and forget about it. If it’s something else, I’ll throw it in my virtual “messy drawer” which is just a folder in Google Drive.
You are getting quite a few clever ideas in this thread, but I suspect the vast majority of people out there in the real world do more or less the same as I do (of course, I’m biased).
My “philosophy” (overstatement) is that 99.99% of the time I’ll never need it. If I ever do, it might be a little painful to dig around, but that doesn’t justify having a whole system in place. I would rather keep my mess.