Make the broth: In a large pot, add water and chicken on medium heat and bring to a boil. The chicken boiling will cause a foamy film at the top–scoop this out as it cooks.
Add in carrots, potatoes, celery stalk base, bay leaves and salt and pepper. Reduce heat and simmer for about one hour, stirring occasionally.
Remove the vegetables, chicken and bay leaves onto a plate. Let the chicken cool some then shred it for the soup. Make sure as much of the foam is off the top of the broth as possible at this time.
Add the soup vegetables, shredded chicken, and barley to the broth.
Cook covered on medium, stirring occasionally, for about 30 minutes or until vegetable and barley textures are to your liking.
Don't waste the carrots, potatoes, and extra chicken breast leftover from making the stock; these make a great, flavorful snack or use the chicken on a sandwich!