I have a 6KW Gillette diesel generator, (not autostart though it is electric start) and it can pretty much power my house just fine. It's hooked into the panel with a transfer switch. I have hot water natural gas boiler for a furnace, natural gas hot water heater, and city water. So, there's not really any one thing that takes a HUGE amount of power to run.
It was a $4000 dollar generator 20 years ago, but I got it in an estate auction 19 years ago with 2 hours runtime on it, for $1000. It now has about 160~ hours runtime on it, so it'll finally be broken in, in about maybe another 5 years or so, lol.
P.s. At my previous house this same generator ran my 220 volt well pump, water softener, and forced air gas furnace, and every else just fine with no issues. Oh, though I never tried it with the Central air also running with the rest, as that would just be stupid when fans work for emergencies.