Going to need more details.
Essentially what you have asked is "How much money in gas will it take me to get across country?"
You haven't mentioned the distance you intend to travel, the size of the engine, the weight of the car, gas prices along the way ... and so on ...
About the best I can do (or anyone here for that matter) with the info given is this:
The amount of continuously "reusable" capacity in a battery bank is generally designed into the system at about 20% of rated capacity.
Too small of a bank, and you will regularly take it down too far, shortening it's service life.
Too large, and you'll have trouble charging it, ultimately ending in them sulfating to death from chronic undercharge.
300W inverter, assuming 60% efficiency for the cheapy MSW, you can expect to be drawing 500W to run it at full power.
Therefore, to run a 300W load, again assuming the battery is 12V (as is the case for the cheapy MSW), you will be drawing ~40 amps.
The average car battery is approximately capable of 60AH capacity. Not that they are suited for this use, but as an example (for size purposes)...
The harder you pull on a battery, the less usable capacity it actually has. But let's assume a 'perfect' battery here.
20% of 60 is 12, so your 20% window in your 60AH battery is 12AH.
At 40A, this will last just a touch longer than 15 minutes.
Reality is, due to that whole 'harder you pull' thing, at that rate, it would be closer to 5 or 10 minutes before the 'shut me down' buzzer started going off in the inverter.
LVD would probably happen before the 15 minute mark.
Hope this helps...