Yeah, a barn cat is your best bet, esp. as long as they are actually mice and not rats per se. Since not all cats are skilled or motivated mousers, the safest thing is to get an adolescent (or two! highly recommend two) from a litter of someone else's barn cats whose mother is a good mouser. Ask around horse people and feed stores, spring is coming. If you can wait that long
If not, like others have said, look for a feral relocation outfit, or call lots and lots of rescues and shelters and learn to turn a deaf ear to the ones that want to berate you.
The rats/mice/whatnot are there largely because FOOD is there. Make absolutely totally sure you are NEVER leaving EVEN SMALL spills of grain etc. Just a couple pieces of corn is a meal for a mouse.
Can you grain your horses in their turnout areas rather than in the barn? That will help considerably as well. A lot of rodent meals come from what horses spill. It may require you to stand there while they eat to keep them from reassorting themselves among feed tubs if you have group turnout, but truly it goes a LONG way towards discouraging barn vermin. Use those indestructable fortex tubs on the ground, they're great.
And obviously make totally sure your grain storage is 100% rodentproof.
Our current solution up here is wait til the barn nearly floods then freezes (to shoo them out of underground hidey holes) then have a few nights that get down to -20F. Amazing the dent that'll put in the barn mouse population. Probably not going to work for you, however <vbg>
Good luck,
Pat