Th power button on my APC Back UPS ES 650 appears to be stuck and it won't power on

plug something (other then a computer)into it, turn the gadget on, then unplug the UPS. If the gadget continues active the UPS is OK. Generally after you have punched every button (circuit breakers). After that put a Volt/Ohm meter on the batteries (and the charging wires to check the charging curcuit is OK.

the chirp means a power event has occurred and the unit needs reset to continue operation , power events can be a few things ( to large a current was applied as a load to it , the power stayed off till battery failed and it had to charge it back up when power comes back on )

try turning on the power connection

What I have seen with these units is the battery seems to fail in first two years a lot.

It sounds like its possible your battery is discharged or close to the end of its life, and needs to be replaced, but without better details its impossible to know the exact cause.

A user manual for your ES 650 is available here: http://www.refurbups.com/KB-M-Backups

Check the batteries first

try plugging the pc directly into the wall...if it works the UPS is defective.  You can order new batteries for the UPS online.
Battery Planet.com

So you had the system is storage? If that is the case then it is probably time to change the battery. Storing batteries when they aren't fully charged, or in extreme temperatures can damage the battery. You can check out http://www.batteryinternational.com/ups-batteries.html for a replacement .

Check the relay Circuit. the relay switch must be faulty

possibly..you have unpluged or dislodge you power cable from the outlet or from the power plug shuye at the back of your ups.. double check all the plug and re plug all your power cables !!!! make sure they are tightly pluged in and firm in their outlets..