I have a Sears Craftsman 1/2 HP garage door opener. Last week it
stopped working for me. When I turn it on, the motor makes a noise for
about a second, and then stops. The lights and everything else seem to
work. The number on the motor is 04302GP. The wires going to it from
the capacitor and the power supply seem to be melted a little, so I
suspect it has burned out. is this easily replaceable, or should I just
spring for a new garage door opener?

Did you check to see if the door was blocked or locked, or a spring was
broken? Sounds like the motor is either burned out, or cannot overcome load
of door.