Season beef with salt and black pepper; coat with flour, shaking off excess.
Heat vegetable oil in a Dutch oven or heavy pot; cook and stir beef until browned on all sides. Remove beef with a slotted spoon and set aside. Add onion, cabbage, celery, carrots, and garlic; cook and stir until carrots are softened and onion is translucent, about 10 minutes. Stir in mushrooms, rosemary, thyme, and barley. Return beef to Dutch oven and continue to cook and stir for 3 minutes.
Stir water, beer, and beef bouillon into beef mixture; bring to a boil, cover, and reduce heat to medium-low. Simmer until beef and barley are tender, about 90 minutes. Add brown sugar and vinegar. Simmer, uncovered, for 10 to 15 minutes more.