Back it up on hardware you own. You can buy multiple terabytes at low cost, you can use duplicity/Deja Dup if you want secure remote backups (on a server of your choosing), etc. You can send people files without having to rely on any specific service provider -- there is email, there is rsync, etc. It is cheap and easy to set up a file server and dynamic DNS, assuming you have a broadband connection (surprisingly many people do not, but that is a separate issue).
The problem with these web services is that they allow a single service provider to screw you over on a whim. Why put yourself in that situation? Why leave yourself vulnerable to unscrupulous competitors and governments who might try to copy your files without permission, because some service provider was more concerned about complying with warrants and monetizing your data than about security? Stay in control of your data, and you will not be posting these sorts of stories to HN.
I think there is a much bigger issue here than the theoretical storage solutions. Everything you said is indeed entirely possible for most HN-readers and tech people but I sure can't talk my mother into switching to dynamic DNS and rsync.
The only reason there was a happy ending to this story was that this guy was connected enough to know how to contact someone at Google, what about the other 300million users who barely know what OS they are running? There needs to be some kind of restructuring to protect the public, whether its simple awareness or legislation.
This thing happened to my ex, although it's more likely she did the wipe. She had her home machine and two office machines synched with DB, and her stuff was gone at home and Office One. We went to Office Two, disconnected from the net, turned the machine on and copied her data to USB.
I'm running mine off a Synology DiskStation that sits at home. Mobile clients for Android and iOS. Sync clients for Windows, Mac and Linux (plus source code if you want to write your own): http://owncloud.org/sync-clients/
You could point OwnCloud at the same folder as Dropbox. Both monitor the same folder. Double backup.
Any serious amount of data storage requires a hard drive (as opposed to flash), and hard drives are quite fragile. If you drop one on a hard floor it's probably done. And with time they fail for all kinds of other reasons. They are mechanical devices.
I seem to have had better than average luck with hard drives; it's been about 20 years since I've had one die while I was still using it. In any case, yes, I rely on a great big hard drive for backup, but it sits on the bottom shelf of a large cabinet; it's not falling any distance onto anything.
What's more important is that I know exactly where it is and who has access to it, and I will continue to have access to it as long as I keep on feeding it electricity.