The APC SmartUPS RT 5000 which was lent to Planetarium by P has some slightly unintuitive on-off procedure.
When there is mains power:
Switch on - First switch on input power, if not already on. Wait for 10 seconds or so for various relays to click, with the fan spinning up immediately. Only then press the ON button, for output to appear.
When there is no mains power:
Switch on - If the inverter and fans are off, first press (and release) ON button to start the fans spinning. After 5 seconds, keep ON button pressed till it emits a long beep after 4 seconds or so. Release the ON button during the long beep in order to cold start UPS output.
Switch off - Pressing the OFF button just stops the output - doesn't stop charging or the fans. There doesn't seem to be any way of turning off the fans when the UPS is connected to power - probably the charging circuits need the fans. So, if you want the UPS to be quiet and the fans switched off, you need to switch off inputs mains AC power, then long press the OFF button for completely switching off.