It should not be a problem for one hard disk.
You may, MAY, have trouble getting the power supply to run because they (generally) need a minimum load of about 500mA on the 5v rail to achieve a stable voltage regulation. No load on 12VDC rail is (generally) ok.
Good luck.
Also, the power supply may not work if it does not sense that it is spinning. The third wire to the fan (if connected) may have logic to sense if the fan is spinning. Worth trying before worrying. Just note that the supply may shutdown if it does not sense a spinning motor.