Grab some figure for the electrical load, say 100 amps (most modern houses have 200 amp service). 100 amps X 120 volts = 12,000 watts. One HP is 748 watts, so 12,000/748 = 16.04 HP.
Nothing is 100% efficient, so figure more like 30 HP to run the alternator. One man in the 1920's rigged a Model T Ford engine to supply his electricity and hot water needs.
If you buy a generator set, the manufacturer will have a properly sized engine included.
In my house, just the bare necessities to get by include a refrigerator, a freezer, a few lights, and the gas furnace blower for a total of 5kW. Starting the induction motor in the furnace is the most problematic. The ratio of starting current to running current in a fractional horsepower induction motor (like in the furnace, fridge, and freezer) is about 15:1, so it takes large generator to provide the motor inrush current.