Brown oxtails. Generously season oxtails on all sides with salt and pepper.
Heat oil in a large heavy-bottomed pot over medium-high heat. Once shimmering, add oxtails (you might have to do this in batches depending on the size of your pot). Cook until browned on all sides, about 8 minutes total, then transfer to a plate.

Saute vegetables. Add onion, celery and carrot to the pot and season with salt and pepper. Cook until just tender, about 5 minutes. Add garlic and cook 1 minute more. Add oregano and bay leaves then add back the oxtail and any remaining juices on the plate.

Simmer. Add the water and bring to a boil, then reduce the heat to a simmer, cover and cook until oxtail is tender, about 1 hour.

Add barley. When a sharp knife easily slips into the oxtails add the barley to the soup. Cover and cook for an additional 30 minutes.

Serve. Season with salt and pepper and remove oregano sprigs and bay leaves then ladle soup into bowls. Serve with a generous dollop of chimichurri on top.