To answer the original question, a fridge requires quite a lot of power to operate. Could be 500W. There's also power loss from the voltage conversion, so you need a battery and an inverter that are able to provide more than that - let's say 600W. Car batteries are typically 12V lead-acid batteries. 600W means 50 amps from the battery. That's a huge current. Lead-acid batteries can handle high currents for a short period of time, but high currents have a negative effect on the battery capacity. So my guess is that the fridge could work for a very short period of time.
To answer the original question, a fridge requires quite a lot of power to operate. Could be 500W. There's also power loss from the voltage conversion, so you need a battery and an inverter that are able to provide more than that - let's say 600W. Car batteries are typically 12V lead-acid batteries. 600W means 50 amps from the battery. That's a huge current. Lead-acid batteries can handle high currents for a short period of time, but high currents have a negative effect on the battery capacity. So my guess is that the fridge could work for a very short period of time.