In a dutch oven, heat the oil over medium-high heat. Add the celery and onion and cook, stirring occasionally, until softened, about 5 minutes; season with salt and pepper. Add the flour and cook, stirring often, for 5 minutes. Add the swiss chard stems and cook for 1 minute. Stir in the barley and 6 cups water and bring to a boil. Lower the heat, cover and simmer until the barley is just tender, about 30 minutes.
Working in batches, add the spinach, escarole, parsley and swiss chard leaves, stirring the greens until wilted between each addition. Season with salt and pepper.