Some people who don't have room for a large horse are getting a miniature horse. some of these only grow to be 25"-35" tall, so are almost like haiing a large sized dog, and can easily be kept in the back yard as long as you have an adequate shelter for the mini-horse.
Although they are way too small for anyone except a small child to ride on, people do train them to drive, and they can then ride in the cart once the pony has learned to be driven.
Some areas that do not allow horses in the city limits would prevent you from having a miniature horse in the back yard; but if there were not these kinds of restrictions where we live, I would really enjoy just having one to keep as a pet.
Here is a webpage showing some beautiful mini's showing off, annd pulling carts. They are just so tiny and so exquisite !