I recently purchased from ebay an X61 Thinkpad. The first thing I did was order a new battery, because the old battery will only get 20 minutes.

The new battery came and was not detected at all by the laptop. I wrote the ebay seller, and he sent me instructions that were totally useless. Something about calibrating the battery in BIOS.

I updated to the latest BIOS and there is no feature to calibrate batteries, and not in the extended BIOS utility either.

I bought two more new batteries, and the laptop will not recognize any of these batteries.

I've done everything. I upgraded the laptop to Windows 8, then downgraded it back to Windows 7. Downloaded all the Thinkpad updates. I ran the battery reset feature on the Thinkpad power manager. But that only works on the old battery. Like I said above, the new batteries are not even detected and don't appear in the power manager.

I've about given up hope that I can ever use this laptop on batteries only. Is then anything I can do?

You probably bought cheap knock-off batteries, instead of proper IBM/Lenovo batteries.
'nuff said.