I'm going to have to take the "safety police" road here.
If the generator comes with a big tag that says "do not operate indoors", which it probably does...then I'd say "don't operate it indoors". That would include a shed, but not a small shelter that you can't fit in.
If you watch the news after these hurricane hits, a lot of times there's a follow up story about somoene that died of carbon monoxide poisoning in the aftermath due to a generator being operated inside.
If you leave it outside, someone steals it. If you take it inside, it can kill you.
If you can swing it, I think one of the enclosed backup gen-sets they sell at the box stores with an automatic transfer switch would be the best solution. We've got one at work that will carry our whole building. It even comes on every friday automatically and runs a self test. When the power goes out, it comes on automatically and transfers the load in about a minute. When the power comes back up, it waits about a minute and then transfers the load back and shuts down. It runs on natural gas, so it never needs refueled. It makes CLEAN power which our computers and such will operate just fine on. Before we had it, we blew up more than one computer power supply trying to daisy chain a bunch of extension cords and providing power from a small portable generator.
I just took a $500 problem and came up with a $2500 solution. I'm pretty good at that. It's a really good solution though.
Phil