The requirement is somewhat vague. When you say "Backup copies", is that implying something like (say) LTO tape? Or are you keeping your "backups" on disk? Once you reach LTO-4, LTO is incredibly easy to encrypt using native LTO encryption. If you are leaving everything on disk for your "backups" and performance is not a concern then you might be able to use something like LUKS + dm-crypt to make an encrypted filesystem. Or maybe you run a script that does a 'gpg' encryption of frozen data. You really haven't said a lot about your target environment so any one of these is as valid as any other.
One of the important things about planning for encryption of data at rest is identifying what threats you wish to protect that data from. Like in the above examples, in the case of LTO-4 tape the encryption of the tape itself is a powerful way to be sure that if a tape is misplaced it is not trivially read. And an encrypted filesystem with LUKS + dm-crypt is great to protect disk drives attached to servers in the data center from being physically carried offsite and read. But, with the LUKS + dm-crypt approach, the operating system maintains the full ability to read the encrypted storage and decrypt it on behalf of the user. In terms of threat modeling, an OS-level encrypted disk provides almost no protection from a piece of malware that uses the OS features to read the disk on its behalf.
You have to know what threats you are trying to protect your data from before you start trying to pick out what encryption systems are going to work for you.
