Haha I wish I could have a barn cat! I adore cats. When I eventually own my own property, barn cats are going to be an immediate, or earlier, addition. So many barn cats.
RACCOONS! I'm glad someone mentioned them! I hadn't even considered them but there is an 100% chance they could get into the tack room so I probably should prepare for them to do so at some point...
I found an old, empty, giant Rubbermaid tub in my garage so I'm going to give that a try first, then proceed to Plan B (metal garbage can with bungee cord) if the free option doesn't work.
I would do rat bait but since we're basically in the middle of the forest, coyotes come around all the time as well as a variety of rare-ish hawks (including a bald eagle!!!) and I'd hate to inadvertently kill off some cool animal while trying to get the mice out of my grain.
Thanks for all your responses! I feel (and am!) much more informed. :)