Buying the horse and renting the stable space would be crazy expensive. It's a nice idea, and I'd like to se it happen, but that's a longer term idea.
Stable space honestly isn't that bad to rent, if you're in a country that's got the space for it/a horse culture already in place. My sister did it for a few years after my parents' moved to take care of her old horse in his dotage, and she wasn't exactly raking in dough. It's a few hundred bucks a month, so not cheap, but if you prioritize it, definitely doable for middle class folks.