EyeTV 1.8.4 doesn't provide a simple way to back up your recordings. This program allows you to archive your EyeTV Programs to DVDs.
Once you've done this, all of your recordings disappear from the EyeTV Programs window; when you mount a DVD and restart EyeTV, the programs on that DVD reappear in the Programs window and can be played directly from the DVD.
This is a very simple-minded program; it simply walks through the recordings and backs them up in the order it finds them. A more flexible version (with a better interface) is forthcoming.