It is possible to do an excellent job keeping horses on small acreages, but the *key* thing to remember is the smaller the acreage, the more labor and cost intensive. If I were going to keep a horse on less than an acre in a arid climate, here's a couple of things that I would consider "must haves" -
Excellent quality fencing. Someone with good grazing and companion animals might get by with electro tape, but smaller acreages require excellent, secure fencing because the animals are much more likely to challenge the fence.
Cross fencing. You need to divide the are into 2, 3 or even 4 grazing areas to manage the mud and mess. One of the areas needs to be a "sacrifice" area that the horse goes out on in bad weather, the others need to be managed carefully to maintain some limited grazing.
Manure removal or composting. Even if you know a very serious gardener, on 1 acre, you will need a very well maintained manure compost pit or regular manure removal. The smell of the manure composting other than in a well maintained pit will be ... interesting.
Reliable sources for hay and bedding and storage for both. This is a tough balance. You can either pay more for regular small deliveries of both products, or sacrifice some of your precious space for storage.
Another area close by for exercise, or a truck and trailer to haul somewhere for exercise. A horse will not get its basic exercise needs met wandering a 1/4 or 1/2 acre paddock.
A tiny companion animal, like a mini goat. Most horses just don't do well by themselves, while there are exceptions, I would try to find a tiny, low maintenence companion animal.
For me, in the SE US, it made much more economic sense to buy a larger property and have lower horse maintenence cost and labor. But that's me, and the SE US - your mileage will definitely vary.