Heat the oil in a large saucepan over medium-high heat. Add the turkey and cook, breaking it up with a spoon, until browned, 3 to 5 minutes; transfer to a plate.
Add the onion, carrot, and celery to the saucepan and cook, stirring often, until softened, 3 to 5 minutes.
Return the turkey to the pan and add the broth and barley; season with ¾ teaspoon salt and ¼ teaspoon pepper. Reduce heat and simmer until the barley is tender, 25 to 30 minutes. Stir in the spinach.