So “qué barbaridad” means something like “what a barbarity!” and would be used when we would say “how terrible!” or “oh my god” in response to a bad event that has just taken place, such as a natural disaster.
Example: “¡Japón acaba de sufrir un terremoto terrible!” = “Japan has just suffered a terrible earthquake!” “¡Qué barbaridad!” ¿Cómo no?