At a guess I'd say there is a problem with the PSU, "you can't turn the psu on if it is already on". Is there another supply to the PC maybe an external usb supply is back feeding and running the fan telling the PC it is still running.
Very odd condition and hard to evaluate without seeing it.
does the PSU have an external switch, if so try turning it off there.
Another thing to try is cleaning power supply plugs with electrical cleaner in case of a bad connection. Not sure what else to suggest.
Good Luck.