I think I'd build a roof on 4 short legs to keep snow of my back (and the generator). If it gets windy a sheet of visquene would break the wind and make my life easier getting the thing started. You might even find something at the big box store like a blow-molded plastic enclosure that would work.
I don't like the idea of storing a large amount of fuel for a long time either but on the occasion where I might need 10-20 gallons 'just in case' I would just dump that in the truck's fuel tank after the need went away and keep a smaller amount on hand until the next time.
Maybe you could start the generator up to get the house nice and toasty and then kill the power until it gets cool inside again. That way you wouldn't be burning fuel continuously.
