Heat oil in a large saucepan over medium heat; add onion and celery and cook, stirring, until beginning to soften, 2 to 4 minutes. Grate or finely chop 1 clove garlic; add to the pan and cook, stirring, until fragrant, about 1 minute. Add broth, chicken and barley. Bring to a gentle simmer. Cover and cook over low heat until the chicken is cooked through, about 20 minutes. Transfer the chicken to a plate with a slotted spoon. Return the broth to a simmer and cook until the barley is tender, 20 to 30 minutes.