If your outside condenser fan is running, that eliminates any problems with the thermostat, contactor and safteys. The unit is getting its 24 volt signal and power is getting past the contactor. I would check the following
Bad capacitor, not all capacitors bulge when bad
Burnt wiring at compressor terminals on side of compressor, they make a repair kit for this.
Ohm out the windings on the compressor for a possible open.
Make sure all wiring is good going to the compressor. visually check all terminals and sta con connectors.
Compressor could have been out do to thermal overload. If it starts running after it cools, check its amp draw
EDIT: sorry didnt see your post from last night, if it is running now and quits again, I would check the amp draw and faulty wiring going down to the compressor, still check the connections down on the compressor.