Heat olive oil in a heavy-bottomed pot over medium heat; cook and stir onion until translucent, about 5 minutes. Add carrot, celery, and salt; cook and stir for 2 minutes. Add barley and mix to coat with oil; cook and stir until barley is golden brown, about 5 minutes.
Stir chicken stock, water, thyme, and sage into barley mixture. Cover pot and bring to a boil; reduce heat to medium-low and simmer until liquid is absorbed, about 45 minutes. Remove pot from heat and stir mint into barley pilaf. Cover pot and allow flavors to blend, 5 to 10 minutes.