Ideally you would build a large multi-purpose structure that you can use as an arena but adds to the appeal of the property for many other uses, shop, storage, whatever such someone may want to do with it.
For that, a commercial metal structure builder would work best for your money.
Think about framed clear span all metal barns, the eaves tall enough for any you may want.
Most such here are 16’ at the eaves for plain riding arenas and, since most are used at times for roping, 18’ is best.
That gives you more usable height in the structure because of the clear span framed gable.
Up to 100’ the cost is incrementally small, over that the engineering needs get considerably higher so ever foot wider cost more than you may think.
Example, between 80’ or 100’, every extra foot will cost about the same.
Over 100’, every extra foot may cost 2x more at first, if adding more 3x more, 4x more.
That 100’ cut off is regionally dependent.
That is so here, where you are they may need to start beefing the framing at 60’ or 80’, or 120’.
Ask the building companies you get bids from.
That is important if you have a small, fixed budget and who doesn’t.
Staying under or up to that threshhold if you can may save you thousands.
Our covered arena is all purpose, but mainly for our area a roping arena.
It is 120’ x 250’ and is used for training for ranch horses, cowhorse, reining, all kinds of junior rodeo events, barrels, poles, calf roping for ribbon, breakaway, etc.
What footing you put in there is important, will determine what kind of training and riding you can do there best, so pay attention to that also.
For most you may do, stay with a firm, compacted level base and add your preference, some kind of sand is standard for most such arenas, at a depth of no less than 2 1/2" and no more than 3 1/2".
Less footing depth will make the arena too fast and horses may slip and fall when turning.
More footing depth will demand a bigger effort to get around it for horses and may bring more suspensory, joint and muscle injuries if you push a horse.
Cutting arenas are one exception, generally deeper than any and so are also where you have to train more carefully.
I would go to your local event centers in towns around you, college arenas, to private indoor arenas and ask who build them and if they like them and what would they do different.
Maybe in your area they are best oriented a certain way, so any snow blows by, rather than pile on around the edges?
Or one roof pitch is better than another?
Today most insulate indoor arenas with spray foam insulation, the kind you can paint over with thick glossy paint you can powerwash and birds and insects don’t damage.
It is also the most costly insulation.
Most important, don’t settle for the first idea or builder you find, but spend as much time as you can wait to start by learn all you can about your options, so you build the best that will fit best for your situation and will be flexible enough to serve later as your needs will change, as they surely will once working out of there and down the years.