It's easy to tell if your side-by-side refrigerator is stuck in self-defrosting mode. First of all, the compressor will never come on. The humming sound you occasionally hear while the refrigerator is working is the compressor turning on and compressing the refrigerant gas prior to the expansion for the gas that creates the cold conditions.
If the compressor is not coming on then the compressor is either not working or the defrost system or thermostat is not signaling the compressor to do its job. When the defrost doesn't stop it can also affect the cooling ability of your refrigerator. Since the defrost heats up the freezer compartment it can continue warming the interior of the unit and stop the cold air production entirely.
Defrost Timer Stuck
The first thing you'll need to check when the freezer on your side-by-side is in a constant defrost mode is the defrost timer. The defrost timer regulates the frequency and duration of the defrost heater's operation. If the timer is stuck it may not tell the heater to turn off at all. And if it malfunctions during a non-defrost cycle, then it may cause frost build-up. If you have the manual type defrost timer you can turn it and the refrigerator will kick on if this is the problem. This means the timer simply needs to be replaced. If your refrigerator has an electronic timer then you won't be able to manually check it, but it could still be the problem and will still require replacement. The timer is generally in the fresh food compartment or behind the kick plate at the bottom of the refrigerator.
The defrost heater is the key component in the system that melts away the frozen condensation from the coils. When this component malfunctions, the defrost will either not work at all or will stay on permanently. Burned out heaters will result in no defrosting, while the heater remaining on constantly indicates a problem in the wiring or another component. Have an appliance repair person check out this component if other troubleshooting comes up empty.
Another part that could prevent the defrost cycle from stopping is the defrost thermostat. The defrost thermostat, or bi-metal switch, measures the temperature in the freezer compartment to determine when the temperature is too low and needs to be defrosted. If this component is not working correctly it could misread the actual temperature and send a signal for the defrost heater to turn on when it shouldn't.