Coleman mach 8333a871 - I think my start capacitor has gone out. How do I check it, and can I replace it?

Normally a good indication would be that when the compressor tries to kick on, it gives a "moaning" like sound, power draws down and normally will trip the circuit breaker in panel. Yes it can be changed "very carefully" , as they are a dangerous source of electrical shock. Have to remove cover on roof, and you'll find small metal cover to remove to expose the capacitors etc. Look for burned wires as well.

You can check with a multimeter that has a micro Farad checker you can read on the cap what the mF is and what the range will be plus or minus a certain percent. You can change it just remember how you took the wires off. example 80mf cap plus or minus 10 percent would be72-80 would be good cap.