ClaraN, you'd want a solid structure of some sort that would be safe for minis. A lot of the small storage buildings aren't very strong. Even a mini horse could kick a thin shelter and injure themselves.
A mini could work in a big back yard, but keep in mind that horses do best with another of their kind and two minis would turn a back yard into a dry lot and/or a mud pit pretty quickly. You'd have to consider manure removal, too. If you have a compost heap that might work, but I think some suburbs that allow horses require the manure to be removed from the property altogether. A friend of mine used a dumpster and had a refuse company remove it when it was full, but some refuse companies aren't doing that any longer.
I like to give my minis as much space to roam as possible, without affecting their health. I have three paddocks. All total the mini area is about 2 acres. One mini has a smaller dry lot, I have a front paddock with minimum graze and then a larger area for graze and play.
You will need dry storage for feed, hay and bedding. My three minis go through about 1/2 a large bale of hay every day during the winter. I store 50 bags of shavings in the barn for bedding and I keep the grain in my house to prevent spoilage and rodent issues.
I don't want to scare you or deter you from getting minis, but it's important to know what you might be getting into.
If you really want to check into it a good idea is to contact your town for zoning rules and regulations. One of my friends has an acre of land and the town wouldn't even allow her to have chickens. The minimum is five acres for that, which is kind of nuts.
