No such thing as a Barn Owl big enough to get a barn cat. They weigh less than 2lbs. Barn owls will kill and eat more vermin than any barn car will, though. They prefer human built structures for nesting.
The Barred Owl is a bit bigger than a Barn Owl- a little over 2 lbs, but they prefer swampy areas where they can catch not just mice and rats but crayfish, frogs, snakes, etc. They will nest in a box but usually find their own hollow in a tree, or duke it out with a squirrel or hawk to get their nest if they cannot find a hollow tree or abandoned nest. Barred owls are the ones that say “who cooks for you, who cooks for you-aaaaallllll.” When nesting they raise hell and sound like something from the deepest darkest jungle.
The only owl that is going to tame a shot at a cat would be a Great Horned Owl a/k/a the “hoot owl”. They eat bugs, frogs, mice, like other owls do, but also squirrels, skunks, and in the north where these owls are bigger, sometimes even porcupines, if easier prey is scarce. They weigh 3.5 to 4 lbs and have a wingspan of 3 to 5 feet. Females being much bigger than males. Snowy owls are slightly heavier but don’t come too far south most of the time. Great horned owls often nest in trees that have been topped, in hollow trees, in hawks’ and eagles’ nests, and in rare cases old barns or human structures.