So, you're thinking of getting a horse, but not entirely sure what facilities you really need to keep one at home. Here are some things you should think about:
1. Turnout. Now, many people keep horses without turnout, and in some parts of the country land prices are such to preclude sufficient turnout (southern California is particularly notorious). But unless you happen to live in that kind of community and really can't afford it (I'd rather see horses live in a mare motel than end up on the truck to Mexico for want of homes), your horse should have turnout. The general rule of thumb is one acre per horse, but ponies may manage on as little as half an acre. In some areas, such as the south west, you may need as much as five acres to keep a single horse.
2. Proper fencing. Barb wire or bob wire is not proper fencing for horses. Cattle will feel barb wire and back off. Horses will lean on it and injure themselves. If post and rail is too expensive, then hot tape (more visible than hot wire) is the best fencing to contain horses. Never use sheep mesh with horses - their hooves can easily get stuck in it.
3. A reliable water supply. This does mean that IF you get all of your water from a well in a rural area you MUST own a generator to power the well pump. Horses require 8 to 10 gallons of water a day. This should be provided to the horse in weighted buckets or a proper stock trough - not an old claw foot bath tub (Yes, really, I've seen this).
4. Some kind of shelter. You don't actually need a proper stall to keep a horse - a run-in shed is sufficient. Many working horses out west and quite a few lesson horses in parts of Europe happily live their lives without ever seeing the inside of a stall. It's perfectly fine to keep the horse turned out and just picket it while grooming and tacking up. Much healthier than the reverse. However, you will need some kind of horse-sized run-in, no matter what the climate. Horses can handle a surprising amount of cold, but don't do as well with heavy rain and also appreciate shelter from the sun.
5. A place to store hay. Hay must never be stored open to the elements nor should it be stored next to your home or in/next to a barn horses are kept in - hay can spontaneously combust, especially if it gets damp.
6. A system for disposing of manure. Manure can also spontaneously combust. You can compost it or give it away to be composted, but it should not stick around on the property.
7. A companion. Horses are highly social animals and should never be kept on a property alone. If you only want or need one horse, then consider adopting a non-rideable horse from a rescue, acquiring a miniature horse (or donkey or mule) or a goat. Goats also help with pasture management and make surprisingly good pets.
8. A trailer or access to a trailer. Even if you never take the animal off the property you should either own or have easy, rapid access to a way to transport your horse. Otherwise you will regret it if you have a veterinary emergency.
9. Somewhere to keep tack. You can just portion off part of the run-in or use the back of the garage. Although keeping tack in the house is possible, your guests and family members will appreciate it if the smelly horse stuff is somewhere else.
10. Human and equine first aid kits, readily accessible and clearly marked. Either mark both with a red cross and then write "HUMAN" or "EQUINE" clearly below it or mark the human first aid kit with a red cross and the equine with a blue cross.