Here is what I would do:
I would check to see if the compressor and condenser fan motor is running.
If the compressor was hot to the touch that would give me a clue.
That would tell me the relay is stuck in start mode causing the compressor to get hot and quit.
I would remove the relay and put my HS410 start pack to see if the compressor comes on and I would check if it was over amping. The amp draw should be around 1 amp. If it is running 7 or 8 amps then the compressor is shot will over heat and die.
The fan motor by the compressor should run when the compressor does. So if the compressor is running but not the fan then you need to look at buying a fan motor. If the fan is running but the compressor is not then you need to buy a relay. If all of that is ok I would next listen for the fan motor blowing in the freezer, no fan no cool. Lots of times fans will make a racket if ice begins to build in the evaporator.
They will get louder and louder. Then the ice will build to the point the fan won't blow.
I would need to check for a bad defrost heater or terminator if that is the case. Do this test: unplug the ice box and let it sit for 3 or 4 days till it completely thaws out. Then plug it back in. If it begins to work ok then you KNOW you have a defrost problem. Replacing the defrost thermostat and heater is not very difficult, you still need to let it thaw out to do it. Even if you called me out there and I said it was a defrost problem I would still have to leave and come back 3 days later. If you suspect it is the defrost then go ahead and clean out the ice box and let er sit for 3 days. Then until you can fix it just unplug it once a week like grandma did back in the 40s before auto defrost. That's all the smart pills I have. Post a model and I could be a bit more specific because you could have adc
or a timer a control board instead of a mercury based control and in that case their is much more to check. For instance if you had an old schoo timer,
it might be stuck in defrost. It might just take moving that timer screw forward to get it unstuck. Don't know until I see a model number. More here