Hi,
I just had a use case for encryption. I decided to use
dm-crypt to create an encrypted container. It's fairly easy. You may just have an encrypted container for your live data and another for your backup. Once it is open, you can read from and write to the filesystem. Many rescue distributions support dm-crypt out of the box(grml, sysresccd, knoppix).
Interesting would be, how you securely automate that, because a backup that's not automated is worthless for me. And if you do not do it securely, encryption makes no sense in my view. Maybe you can place a pendrive with the key on it in your computer, so it only boots up when the pendrive is there?
Here's a tutorial for you to read(use google for a lot of resources on the dm-crypt topic):
How To Use DM-Crypt to Create an Encrypted Volume on an Ubuntu VPS | DigitalOcean
Interesting would be, what the nature of your data is and what confidentiality level of your data is, so I/we can better understand your situation and maybe help more.
regards,
stomp.