For contineous use you would be using a lot of gas for a little power. To acheive any decent power output the inverter would have to be wired directly to battery and ground and the engine run at a fairly high speed so as not to discharge the battery. I have a 300W inverter in my truck, plugged into a cigarette lighter outlet and use it to power a laptop and a 10" high velocity fan mounted in window pass through between truck cab and camper body, to put heat and ac in back for the dogs( It's too hot in a Texas summer to put them in back without some cooling on long trips). This inverter is about 3.5-4" hx 10"wx 15'L and has it's own built in cooling fan. I can only imagine the size and weight of a larger output model, probably would take a lot of space.
My portable generator, and thats a laugh, a 11Kw continous output 20 hp Honda, weighs just over 400 lbs and burns 1.7 gal /hr and will power my all electric house with the hot water tank turned off. This does include a 4 ton heat pump, refrigerator and two freezers. We can use use any lights, microwave, stovetop, and I once had to start my 220v 3hp table saw, but that was an extreme starting load, everything slowed down/dimmed but it carried it. I will never do that again unless I make sure the wife is not cooking as she was that time, and will turn most other things off.
Even with the high cost of gas I had 50 gal on hand for Ike, and only used 5. You do not have to run a generator 24 hours as day. We usually ran just before going to bed, and after getting up, to cool house down, for cooking, and a few times through the day to keep refrigerator and freezers cold, and kept things comfortable. I did hear some folks running small units constantly just to power a refrigerator and freeze
Bottom line, get a generator if you need emergency power, get one large enough to carry starting loads of everything you might need to have running at once, and consider setting your house up as I did. I installed a double throw switch and installed a 50amp inlet plug for the generator. The switch is between the meter and circuit breaker box. One motion switches off from utility power to off then to turn on the generator inlet, so there is no possibility of feed back to utility lines and harming some utility worker in the area.