Locate the nameplate, and all the pertinent information relating to it's power consumption (generally listed in watts) will be there. Check your owners manual as well, it may be listed there. Or lookup the brand/model info online. Your inverter size depends upon how much wattage is required by the refrig you have.
Unfortunately, inversion is terribly inefficient - A general rule of thumb for figuring battery capacity needed is that the inversion process requires 10 times as much DC power as the AC power you're trying to produce. Some inverters are more efficient than others, which can reduce that ratio, but they begin to get very expensive. While choosing inverters for comparison check the manufacturers specs for their efficiency ratings.
For example, here is a small Danby cube refrigerator: DAR195BL Danby Compact Cube All Refrigerator With Energy Star
The manufacturer claims the energy consumption is 90 watts or 1.2 amps at 115 volts AC power. Using the 10:1 rule of thumb, you need approximately 12 amps of 12volt DC power on an inverter for it to run. Therefore over the course of 1 day, it will use 288 amp hours (12amp x 24 hrs) of DC power IF
it's actually running the entire 24 hrs. There are factors which play into the true usage such as how often the door is opened and the ambient outdoor temperature. If you don't have people opening it up every 5 minutes obviously it will use less power.
Ok so using a base of 288 amp hours daily you need a battery configuration that will power it. The most common and least expensive setup is using 2 GC-2 type 6v batteries in series, which gives you 12 volts and approximately 220 amp hours. Using our numbers above, as you can see, if the refrig compressor is really running 24 hours, you're about 60amp hours short of running an entire day. I have found that the rated consumption numbers from the manufacturers are on the high side and that is it's rating while the compressor is running. For example, that small Danby will most likely only use around 140 amp hours in a 24 hr period depending upon how often it's opened and the ambient temperature.
So to answer your question above, with 2 GC-2 batteries and a fairly efficient inverter capable of producing at least 100 watts AC power, you will most likely get a run time of approximately a day and half on that small refrig depending upon your usage.
Keep in mind all this math is approximate and subject to change depending your
usage patterns, such as how often the door is opened, the ambient temperature outside and the efficiency of your inverter.