Since it would briefly come back to life if allowed to rest for awhile unplugged, I bought a backup hard drive of the proper sort (with its own power supply), and plugged it in. However, the DVR died before backing up commenced.
This DVR is leased, so Dish sent along a new one, asking that I return the broken one within 10 days. Yesterday a technician showed up to be sure that everything was working, since the new machine had taken quite long to get a watchable image. I asked him if it was possible to move the old hard drive into the new machine and do the backup, before sending the old machine back? He said yes, although he couldn't do it for us.
So, new challenge: remove the old hard drive from the old DVR. I found a wikibook about the DVR here: en.wikibooks.org/wiki/VIP_922/Dish_Network. There the author says,
With no A/C power connected - from the old 922, pull the internal hard drive and set it aside. To do this remove 4, back cover screws (black,) then slide the cover back about 1/2 inch and tilt upwards to remove.Well now, here was my first problem. Screws, no biggie. But "slide the cover back"? It simply would not move for me. However, teamwork to the rescue. My husband Bob used the straight-slot screwdriver and pried with a bit more power than I would have used, and slide, it did. From there on out, the problem was redimensionated (thank you genii for that beautiful term!) and it was only a matter of more screws, unhooking the power and motherboard connection, and sliding out.
After using my husband again to help me slide out the TV cabinet and photograph and then remove the DVR hookups, I used the same procedure again. Remove the cover, then the HD, and then switched the old HD into its place. I left off the cover, and then Bob hooked up the new machine again, and we turned it on to wait. After the backup hard drive was plugged in, this slow beast restarted again, but we had a new option: backup. Just to be safe, we selected only the Doctor Who eps. If there is room once that's done, I'll select the rest of what I want. The backup is now proceeding, and the readout reports that it will take another 10 hours. OMG, usb is slow!
So, redimensionating is cool. I'm going to try to remember to do it more often. Also, many thanks to the dish tech and wikibook author who both shared their information freely, and my husband who supplied support, muscle, and didn't give up!