Place chuck roast in a slow cooker. Cook on High until tender, 4 to 5 hours. Add barley and bay leaf during the last hour of cooking.

Remove meat; chop into bite-size pieces. Discard bay leaf. Set beef, broth, and barley aside.

Heat oil in a large stock pot over medium-high heat. Sauté carrots, celery, onion, and frozen mixed vegetables until tender, 5 to 7 minutes.

Add water, stewed tomatoes, beef bouillon cubes, sugar, 1/4 teaspoon pepper, and beef-barley-broth mixture. Bring to boil, reduce heat, and simmer 10 to 20 minutes.

Season with salt and pepper before serving.