Heat oil in a large stockpot over medium heat. Add celery, carrots, onion, and garlic; cook and stir until slightly softened and fragrant, about 5 minutes. Add turkey carcass, salt, bay leaves, peppercorns, rosemary, and sage; cook until fragrant, about 5 minutes.

Pour in enough water to cover turkey carcass and vegetables; bring to a boil. Cover the pot, reduce heat to low, and simmer, skimming fat occasionally, until flavors are blended, 4 to 5 hours. Strain stock into a container; discard carcass and vegetables.