Well that's a new one. It's pretty cut-and-dried: no user serviceable parts inside. If the power LED is off, then it's a power problem.
Either the "wall wart" power supply is flaking out (internal component failure or loose/damaged wires in the cord or plug), or the OBi itself is failing. If you have another 12VDC power adapter, I'd try that next. It's a common adapter used by lots of electronic gizmos like home routers, switches, etc.