Lot size suitable for the number of horses you plan to have
I've been on 1/3 acre lots that felt huge, and flat 1-acre properties that felt unusable. Size might not matter as much as you think, BUT...
Be aware of the zoning for the property and lot square footage. Many horses that have horses on them are not technically horse property, based on current city/county regulations. It's also not unusual for people to have more horses than are allowed for their lot size; don't assume that you'll be able to have as many as the current property owner.
After you've made the financial and emotional investment in your new home is NOT the time to have Code Enforcement cite you and tell you to reduce the horse count.
Good access for feed deliveries, horse trailers, etc.
It just amazes me when I see a horse property that's been developed in such a way that you can't even get a feed delivery past the front driveway. Actually, some "horse properties" that I've seen are set up so that you can't get reasonably get a healthy horse on or off the property. I don't want to dwell on the downside of horse ownership, but try to imagine getting a vet to your horse in an emergency, or, worse, having a horse die on your property and needing to deal with his body in a practical manner. If you can't plan for this possibility, you might want to continue boarding in a professional facility.
Even if you don't have a horse trailer now, plan for the future. There should be a place off the street where you could park a horse trailer. Ideally, it would be behind your gates for security purposes, even if you have to pull it out of the yard when you load horses into it.
Layout of Facilities - stalls, shelter, feed storage
Consider the ease or difficulty of making changes to the setup, if it isn't ideal for you.
Can you feed the horses from outside their stalls? Do you have to walk through one horse's stall to get another horse out? Is there a turnout area, and how do you get your horses to it? where will you wash/ groom/ saddle your horse? Where will you keep your tack? Picture yourself feeding in the rain, in the dark, etc. What sort of terrain will you have to deal with to care for your horses? Dot't forget to plan for manure storage and removal.
Will you have "horsey" neighbors somewhere close by? Don't underestimate the value of a support network! Where are the trails? Is there a public arena? Where can you shop for feed?