The App View frontend is open source: https://github.com/bluesky-social/social-app

Much of the backend is open source as well: https://github.com/bluesky-social/atproto/tree/main/packages

What is not are the extra services they run to provide a better and faster UX. Even if it was open source, it likely costs 10s of thousands to run per month (they have moved largely to "onprem" hardware instead of the cloud aiui)