Could I paint a dog house and storage shed with one gallon of paint or do I need more?

I would buy two gallons. One would cover it, but you probably want to put at least two coats of paint on it. I am assuming the shed has a weatherproof primer or other paint on it, and also assuming it is average depth for an outdoor shed.

Exterior or interior? Going to scrape/sand/prime it first? 1600 square feet doesn't give much information...a house with 10' or 14' ceilings (= taller walls = more surface = more paint) will need more paint than a house with 8' ceilings. For the interior, measure the height and length of all the walls, 'rounding up' to the next foot: height x length = square feet of wall surface. Length and width measurement of the floor will give you the square foot surface of the ceiling. Add up all those figures, that's your interior square foot surface area. Then go to a paint store and look on the cans of paint: it will have a section on the label saying "Coverage," which will say something like "A gallon covers 200 square feet" depending on the paint. Talk to the paint store people, give them your square footage figures, tell them what you're painting, ask for tips, recommendations, things NOT to do, etc. Armed with paint rollers, brushes, buckets, masking tape (I prefer the 'delicate surface' type...doesn't peel the paint off of painted walls), plastic tarps, etc., etc., have fun.