Have you considered an LPG (propane) fueled standby generator, like Generac? They have units that can run on either NG or LPG and if you can get there every once in a while to refill the LPG tank, it may work. Alternatively, if your tank is large enough, you may be able to enter in a contract with a local propane delivery company for them to come fill it up (very infrequently, so I'm not sure if they would agree, but probably won't hurt to ask). Generac's smallest 7,000W unit (<a href="
I don't know of any battery-based backup that is able to run space heaters for any considerable period of time. They are simply too expensive to use for space heaters if you try to size them properly for that. Problem with solar power, of course, is that heating is most needed when there's no sun, so batteries are required and we've back to the beginning of this paragraph.
Also, regardless of the fuel type, a generator is a pretty complex mechanical system that needs to be maintained. It's not something that I would simply leave in place to run on autopilot indefinitely - there would have to be some scheduled visits to check on the thing to make sure it's still alive when it's needed.
The problem of cooking is actually the easiest to solve here - get a propane camping stove like this one