Put mushrooms, carrots, onion, and broth in a 4-quart slow cooker. Rub turkey with oil, sage, thyme, salt, and pepper; place on top.
Cover; cook on high 2 to 3 hours until each tenderloin reaches 165°F at thickest part and carrots are tender. Start checking turkey after 2 hours.
During the final hour, boil barley and 3 cups water in a saucepan. Cover; simmer on low 50 to 60 minutes until chewy-tender. Drain excess water.
Transfer turkey to a plate; rest 10 minutes. Bring vegetables and cooking broth to a simmer in a saucepan.
Mix cornstarch with 2 tablespoons cold water; whisk into broth. Simmer 2 to 3 minutes until lightly thickened.
Slice turkey across grain; divide with barley and vegetable gravy among 4 plates.