You should be able to control all data produced by your phone and be able to collect it all into one place (cloud / p2p or local).
SMS Gate app backups all your SMS from Android/INCOMING folder to the IMAP/INBOX folder of Gmail or your provider of choice (you can setup one in Settings). Once uploaded to an IMAP server you can even reply to the SMS using standard reply e-mail mechanics.
Saving to a local folder allows you to backup it using an app like Syncthing.
Although APK downloads are available below to give you the choice, you should be aware that by installing that way you will not receive update notifications and it's a less secure way to download. We recommend that you install the F-Droid client and use that.Download F-Droid
This version requires Android 2.3 or newer.
It is built and signed by F-Droid, and guaranteed to correspond to this source tarball.