The first is to have your own offsite storage that you back up to, where the backup is (at least) as large as the original. Multiple people have recommended Crashplan, and that's certainly a viable option. There are undoubtedly other options that could do similar things depending on how down into the weeds you want to get - rsync, the various rsync-based versioning backup solutions, git-annex as mentioned by someone else though that one's new to me. I'll note that from experience with Crashplan's Enterprise product on some older 32-bit servers, the client software can chew some fairly significant memory when you have a lot of files or data.
The other and probably simpler option is that when you start to near capacity on the storage system, don't upgrade it - shut it down and store it, preferably not in the same (not-yet-burning) building after building the new system and copying the data over to it. After you shut the old one down, keep backups of anything you've changed since that "checkpoint" system; hopefully your data isn't changing that rapidly - 20 TB seems to me almost guaranteed to be mostly static.
If you do need the paid version, Tacit Dynamics (tacit.dk) apparently has it available through Amazon.com and through AndroidPit.com as well if either of those will work for you. The drawback of either is that I believe (as with most competing app markets) you have to keep that app market installed on your device.
If that doesn't do the trick, I find that my tablet running Cyanogenmod 10.1.3 has rsync 3.0.7, but I've also installed extra bits and pieces ("Android Terminal Emulator" by Jack Palevich, "Terminal IDE" by Spartacus Rex, "Busybox Pro" by Stephen (stericson)) so I'm not 100% sure that it was originally available. You may be able to script and schedule something of your own based on scripts you use elsewhere, though with a few changes (e.g. "jping" instead of "ping", see Issue 29 on Terminal IDE's code.google.com page though that indicates that it may be fixed).
There's also a "lite" version, which only allows 2 accounts, no Tasker support and no sync filters (which I've never fiddled with anyway, so may not be that important).
Copilot is free to use on weekends (their "day pass" pricing is $5 on weekdays and free on weekends).
I also saw it quoted elsewhere originally, so I didn't see the headline at the top.