Check to see if the thermostat's temperature setting dial accidentally got turned way down so that the thermostat is setting is what's keeping the fridge warm.
Is it a frost free compact fridge? If so, the defrost timer is probably stuck in defrost mode. In my view, for the $5 a new one costs, it's a good idea to replace the defrost timers in frost free fridges every 10 years as preventive maintenance.
If it's manual defrost compact fridge, check that there's power in the receptacle it's plugged into. If it's getting power, then it's just a matter of checking for continuity across the thermostat and checking that the compressor motor is getting power.
If the compressor motor is "humming" then it could just be the start capacitor for the motor. If it's not doing anything at all, then it's probably just not getting power.