A lot depends on the type of airplane, redundancy of systems, and the emergency procedures section of the specific Flight Handbook. A look at the MEL/DDPG would give some guidance, but would not likely be deterministic once airborne.
That said, I would likely continue if the airplane was otherwise unencumbered with related deferred maintenance items (e.g., 1 pack already failed) and there was no specific requirements for immediate landing in the non-normal checklist.
"No O2" in this case merely means the pilots' emergency O2 bottles have been depleted. There will still be O2 for the pax, because they are on separate systems. There is also significant O2 available in walkaround bottles and (possibly) in emergency medical O2 bottles. I would ensure there was an adequate number of these installed in the cockpit or brought to the cockpit in case of a fire or depressurization. AFTER that, a call to Dispatch to ask what else they wanted me to do, assuming I was still prior to ETP.
Descending to 10,000' when cabin pressurization is functional is NOT a good idea...