Compressor won't start up ...Ideas?

Unfortunately this can be a number of problems.........start cap, start switch (centrifugal), start relay if used (works on current and or voltage to delay start cap) start winding damaged/ burned / partly shorted.

"What I'd do" likely. If you can't test the start cap, and that can be difficult, just replace it. Take the thing apart and examine the mechanics of the start switch........is the spring operated collar on the shaft OK and free? Not sticking? Is the switch in good shape, the contacts OK? How about SMELL? Does the motor smell like or the windings LOOK like they are burned?

And as others have said, make SURE you have an outlet/ extension with enough capacity (ampacity). The problem EG with long cords is that they drop the voltage down, and don't "give" enough to start the thing.