Since it appears as if you are unfamiliar with the types of homes that are here in The Villages, you might want to look at this web site to see what floor plans are in each category.
Homefinder - The Villages(R) Homes and Villas for Sale
From your description of what you are wanting to rent, it sounds like either a PATIO VILLA or a COURTYARD VILLA would work for you. This web site will show you those layouts so you can determine if it would work. As others have said, the Courtyard Villas have a fenced back area while the Patio Villas do not. But almost all have a 1 car garage that you could use to store your items.
Good Luck!