A milk paint color
Barn red is a deep reddish brown. It comes from a list of milk paint colors. These paints copy the colors once used on the American frontier, and like the old paints, they are made with milk. Painters mix barn red with white to get lighter shades of it. You can try that on screen with the color mixer.
This site uses #7C0A02. The Wikipedia page on shades of red gives #7C0902, which is one step lower in green. Your eye can't tell the two apart.
Why so many barns are red
Nobody kept a firm record of how the custom began, but the usual story is about cheap materials. Farmers sealed barn wood with linseed oil, often mixed with milk or lime. They added rust, which is iron oxide, because it was easy to get and helped keep mold and moss off the wood. The rust turned the mix red.
Later in the 1800s, red paint became the cheapest kind to buy in the United States, so the habit stuck. Some people say farmers also mixed in animal blood, but there is little proof of that. You can read more in Bob Vila's article on red barns.
For a lighter color made from the same kind of iron pigment, look at Indian red. More reds are on the shades of red page.