Apologies for resurrecting an old thread but I experienced the same issue with an HP 33120A today and was able to resolve it painlessly.
I just purchased one of these units on ebay and it arrived today. On first power up, everything worked but the 'Err' indicator was lit. Digging into the menus revealed the single error code 771. Unit passed the self-test with no errors. After finding this thread and digging a bit in the service manual, I was investigating the possibility of calibrating this unit. During that process, I stumbled on to a solution by following this process:
1. Under the CAL menu, set the unit to 'UNSECURED' (default passcode is 033120; procedure described on page 47 of the service manual)
2. With the unit now in the 'UNSECURED' state, power it off.
3. Wait a few seconds and power back on.
4. Check the CAL menu and verify the unit is still 'UNSECURED'.
5. Set the unit back into the 'SECURED' state (again default passcode 033120)
6. Power off, wait a few seconds, power back on. No more Error 771!
It's possible your unit may have had the passcode changed to something other than the default. If so you'll have to open the case and short the pads at JM101 as described on page 65 of the service manual in order to bypass the code.
Basically, we're unsecuring to perform a calibration but then resecuring without calibrating anything. Not certain why this resolved error 771 but my theory is that the act of securing the device perhaps recalculates the offending "Nonvolatile cal memory checksum" to a valid value.
YMMV but just wanted to share this as a possible resolution for anyone else who may have experienced this scenario.
Incidentally, I did run through the quick check procedures outlined in the service manual and my HP 33120A appears to be functioning correctly within spec.