I keep horses very cheaply, and I think well.
We own our own place, and are meticulous about maintaining it, so I have good quality pasture year round. Here in central VA, USA, you can maintain a horse on pasture with mineral supplementation very easily. My horses live out, 24/7 and have free access to a barn or shed for shelter.
So, I have two riding horses and a 10 hand pony, and these are my costs -
$500 - $550 per year for hay. That's 12 - 14 round bales, I feed hay from October to February or March. The rest of the time they're on pasture.
$30 - 40 per year for mineral blocks.
$20 for grain. I buy 2 - 3 bags per year, all I use it for is to put something in can to rattle to call the horses in.
$450. - $500. for yearly vet visit, vaccines and coggins. I do their fall boosters myself for another $50.
Farrier every 5 weeks - $75 - $215; depending on who's wearing shoes.
So I keep 3 animals year round for a total annual cost of around $3000.; farrier costs being by far the biggest expense. And they are fat, shiny and healthy. Two are currently wearing grazing muzzles, and the third probably needs to.
But here are the hidden or soft costs to my arrangement - I haven't included diesel fuel, maintenence or initial investment in my tractor. Without the tractor, I couldn't be so scrupulous about maintianing the quality of my pasture, and I couldn't feed round bales and would have to use a more expensive method.
I lime and or fertilize and seed my pastures every other year, or as needed. That can be another $1000. And while I don't muck or feed daily, or bring horses in and out, I spend a lot of time bushhogging, harrowing, putting hay out, etc.
As someone else said, choosing a horse that's an easy keeper and that has good feet is key to keeping costs low. Another key consideration is not having it be stall kept and not having to feed hay year round. A terrific option to full stall board at a commercial facility is to lease a pasture and do the care yourself, but you need enough experience to be comfortable doing the care, and a schedule that will allow you to do so.
I did this for years before we bought our place and it was the only way I could have kept horses; when I was no longer working with horses full time, boarding at a commercial facility just wasn't an option.
Good for you for doing you research ahead of time and really exploring your options.