Two points - just because the DVR and tech support tells you your recordings are not there that may not be true. It may a sign of a failing hard drive, but if you try rebooting (sometimes a few times), your recordings may reappear again. Long enough to transfer. Second, you can't back up as such, but you are allowed to transfer to a backup external drive. You can then restore/transfer back to a replacement dish DVR. (Dish does not let us make backup or copies).
Just saw your post here - Yep no way to get it back - this has happened to me twice!! I got an external hard drive and copied recordings I REALLY want to save.
Yeah but that's no guarantee of anything - I had many many dozens of hours of recordings on an external hard drive, and the bloody upgrade to the new software on the Hopper with Sling did something to my hard drive and the recordings are no longer there.
When I contacted Dish she was very apologetic, said I wasn't the only one this happened to, and that there is likely nothing that can be done to retrieve them.