Heat vegetable oil in a large stockpot over medium-high heat. Add turkey bones to hot oil; cook, turning occasionally, until browned, about 10 minutes. Transfer bones to a bowl.

Cook and stir quartered onion, coarsely chopped celery, and coarsely chopped carrot in the hot oil in the same stockpot until fragrant, about 2 minutes. Return turkey bones to stockpot and add 16 cups water and thyme sprigs. Bring to a boil, skim off foam, reduce heat to medium-low, and simmer until liquid reduces to 10 cups, about 2 hours. Strain stock into a large bowl and let stand for 15 minutes. Spoon fat off top of stock.

Bring 2 1/2 cup water and barley to a boil in a saucepan. Cover saucepan, reduce heat to low, and simmer until the barley is tender, 30 to 40 minutes.

Heat olive oil in a large stockpot over medium-high heat; cook and stir diced onion, sliced carrots, sliced celery, and garlic until slightly softened, about 5 minutes. Add turkey stock to onion mixture and bring to a boil.

Mix turkey meat, barley, parsley, thyme leaves, salt, black pepper, and cayenne pepper into soup; reduce heat to medium-low and simmer for 20 minutes. Stir lemon juice into soup.