"Time to install Windows 8 on this old laptop. Oh crap what will I do with my old docs?"
I finally came across a backup service that was cheap and from a company I can be sure will be around for awhile. I'm talking about Amazon Glacier. You may or may not have heard of it, since it was released just two months ago, and it's heavily developer focused. Meaning there's an API but no official GUI or console.
Thankfully there's FastGlacier which is a free-for-home-use Windows program to make uploading and downloading easy. I should mention Amazon puts a fee on downloads that surpass 5% of what you've stored there; the service is meant for long-term infrequently-accessed archival storage, completely unlike Dropbox.
There are a few command-line programs for Linux. Look on github or this Quora thread. I tried one of them, glacier-cmd, but couldn't figure out how to do directory uploads other than "find | xargs glacier-cmd upload". Maybe I just answered my own question. Oh well. I ended up copying my files to Windows first and then using FastGlacier's nice and intelligent GUI.
The service is extremely cheap for typical home backup. Uploading 1000 files costs 5 cents, with a recurring cost of 1 cent per GB per month. For me, I stored about 20 GB of files, many of them tiny metadata and random small text files. It came out to an upfront cost of $4.08, and I'll pay $2.40 a year(!) to keep those files there. That's literally pocket change. I can do pickup at the Thai restaurant instead of delivery and make that money back and then some.
Compare with another well regarded backup service, Backblaze, which costs a flat $3.96/month for unlimited storage. I haven't used them but I hear good things. I'm sure they include a nice GUI too. But to justify the cost I'd have to store nearly 400 GB first. I'm sure a lot of people do store that much data. Someday when I have Ultra-HD family videos that I want to back up, I can shop around for new options. But by then who knows what companies will still be around; maybe computers will be obsolete and we'll encode all our data in DNA converted to dark matter.