On a mac, when you press and hold the power button, it starts the firmware update process, which causes the power indicator LED to flash rapidly and is followed by a long beep. Since you stated that you are currently experiencing issues with you power button, I would think that it is likely the culprit. The power button is a momentary switch, but it sounds like yours is making constant contact.
However, if this is not the issue, you might be able to use the following reference for apple support to diagnose the problem:
Power On Self-Test Beep Definition
"If a fault is detected during the test, you will not hear a normal startup chime. Instead, the system will beep as explained below. If you experience one of these beeps, you may call your Apple Authorized Service Provider for additional troubleshooting assistance.
1 beep = no RAM installed
2 beeps = incompatible RAM types
3 beeps = no good banks
4 beeps = no good boot images in the boot ROM (and/or bad sys config block)
5 beeps = processor is not usable
In addition to the beeps, on some computers the power LED will flash a corresponding number of times plus one. The LED will repeat the sequence after approximately a 5-second pause. The tones are only played once.
Note: In this case, a flash is considered to be 1/4 second or 250 ms or greater in length. "
If none of the above helps, try removing your RAM module and see if the problem persists.