Adjust oven rack to lower-middle position and heat oven to 450 degrees. Heat a large soup kettle or Dutch oven over medium heat.
Coat both sides of roast with oil and season with salt and pepper. Increase heat of pot to medium-high. Add roast; cook on one side until well-browned, about 5 minutes. Turn and continue to cook until second side is well browned, about 5 minutes longer. Add onion; saute until tender and starting to turn brown, about 5 minutes. Add 2 quarts broth and the barley. Turn heat off.
Using a potholder or towel, to protect your hands, cover the pot with a sheet of heavy-duty foil, pressing on the foil so that it is concave and touches the roast. Seal the pot completely around the edges. Cover pot with lid. Turn heat to high until you hear juices bubble when you do remove pot to oven and cook for 2 1/2 hours.
Remove from oven and let stand, covered, for 10 to 15 minutes. Remove lid and foil and remove roast from pot. When cool enough to handle, shred meat into bite-size pieces with a knife and fork or fingers. Return meat to pot, along with remaining quart of broth (if needed) and simmer until heated through. Serve.