This is a great question/scenario and one I've thought about.

Water is the key.

By definition, you have shelter. And warmth can be found with the tineist bit of fuel from furniture and trees if you make one (small) room warm and bundle up.

Food is a challenge, but you are home so hopefully have some set aside. Plus critters or gardens can provide for quite a while.

So, back to water... The rain collection from down spouts is spot on.

Also if you are fortunate to live in a home with a well you have a very good option. A generator that is only run for a few minutes each day can last for a LONG time. And in that few minutes your well pump can produce gallons and gallons and gallons of water for you.

Normal generator runtime is around 8 hours for half load? And most wells produce more than 4 gallons per minute flow. So say you run that generator for <10 minutes a day to secure fresh water only. You are talking about 8 weeks or more without a refill. And a single 5 gallon can of extra gas/diesel can get you through months. And don't forget... there is gas in your lawnmower, trimmer, and autos.