I save PDFs and ironically I print hard copies of some.

A combination of both. For bill like statements, for example credit cards, I review them as a PDF and then make payment on line. So I simply save the PDF to a folder designated for such. For banking, they tend to hold a years worth online. Anything older than a year and they tend to charge you for it. So for banking and the like, I do those on a quarterly basis at the longest.

Additionally I have two USB sticks. One is primary and one is the backup. I store the statement information on the primary USB stick as described above. I then periodically copy it over to the backup USB stick as a safety. The USB stick approach serves two purposes. One, is portability. The USB stick is not tied to a specific computer. Two, is safety. Let's just say we have a burglary/robbery problem in the Orlando area. USB sticks are small and easy to hide. Computers, laptops, external storage ala NAS, not so much.