Can someone suggest a cheap backup drive that holds maybe 1 tape for backups, is still sold, still popular, and is likely to still be accessible on it or another model in another 10 years?
You don't specify a price range, but your use of the word "cheap" implies you looked at what's commercially available and were shocked by the fact that even low-end tape drives run around $2000.
I'd recommend getting a secondhand Quantum TC-L52AN (internal drive) or TC-L52BN (the same thing, in an external enclosure). I don't have experience with this particular model, but I've had good luck with Quantum tape drives in the past. They run around $500-$800 on Ebay; you'll probably need a SAS controller card as well, but those can be found for under $100. The LTO standard requires two generations of backward compatibility, so the current LTO-7 drives can read the TC-L52AN's LTO-5 tapes. Ten years from now, you'll be able to get a secondhand LTO-7 drive to read your tapes.