When considering a solar system, there are several efficiency factors that must be taken into account. A common mistake is to simply look at the "name plate" power information of the solar panel itself. This is not a sufficient analysis.
Solar panels are rated in watts by the manufacturer. These are the watts that can be achieved under ideal, direct overhead sun conditions. But these conditions do not exist for most of the day, not all days are sunny, and the angle of the sun changes seasonally. To account for these real world conditions, a factor known as PV solar radiation must be included. This factor is specific to the geographic location of the panel and is based on historic solar radiation information gathered from that location. An example of this data for the United States can be found here. Your solar supplier should be asked to quote the annual energy (kilowatt hours) output of the solar panel(s) based on your annual PV solar radiation for your location.
Another important factor to be considered is the inefficiency of the charge controller and inverter system. In the case of an off the grid system, the inefficiency occurs twice - first as the controller charges the battery from the solar panels and again as the controller converts the energy stored in the battery to alternating current. There are also losses in the solar panel wiring and in the battery itself. The combined losses can be estimated at 80% if your cannot obtain specific data for your controller, battery, and site wiring plan.
A conservative plan for an off the grid solar system takes into account the possibility that the battery may be nearly fully drained at the same time there are loads from the household requiring energy. This can occur after a period of inclement weather, for example. So the system would be sized to fully charge a depleted battery during a single day of sun while supplying the energy to the attached loads at the same time.
Your refrigerator, while rated at a maximum power consumption of 300 watts is not the information you need to size your solar system. The real question is how much energy per month or year does your refrigerator consume. This will be far less than 300 watts * 24 hours * 365 days/year. Depending upon your location and the age of the refrigerator, it may include the estimated annual energy consumption on its nameplate. If not, you may be able to look it up here. Failing that, you will need to measure the consumption of your refrigerator over say a one week period. There are various inexpensive devices available on the Internet that perform this function. Here is an example of such a device.
Another factor that must be considered with any motor powered appliance like a refrigerator is the startup current that is required. This is typically much higher than the running current of the refrigerator. So while this has little effect on the overall energy requirements of your panels, the battery and inverter combination must be able to handle this "blip" of higher power consumption while keeping all other connected devices powered. If you are not able to measure the startup current or power drawn by your refrigerator, a conservative estimate is that is will take 3 times the run time current or power. Your battery and inverter must be able to supply this power requirement combined with all other connected loads. This sets the minimum power specifications for your battery and inverter combination.
There are several factors that go into sizing the battery:
1.) The battery must supply some or all of the energy during inclement weather. This means that if you have two very cloudy days in a row, the battery may have to supply all of your energy needs for a 48 hour period.
2.) The inverter effectively reduces the capacity of the battery. A 10% loss from conversion efficiency is not uncommon.
3.) Battery amp hour ratings are based on a 20 hour discharge period. So a 100 AH battery is rated at 5 amps for 20 hours. If you attempt to draw 100 amps for 1 hour, the realized energy capacity of the battery will be significantly lower - probably in the order of 65 AH.
4.) The temperature of the battery has an effect on its realized energy capacity. For a typical AGM battery, freezing temperatures reduce its capacity while higher temperatures increase its capacity. Higher temperatures also reduce the overall lifetime of the battery.
5.) The inverter will drain your battery even when there is no load if you do not turn it off. This minimum load on the battery must be taken into account but do not duplicate inverter inefficiencies in this calculation.
6.) You should always build in some reserve capacity into the battery for unforeseen circumstances. Running the battery at the edge of its ratings is never a good strategy.