BuddyPress.org

@dubbelf –

Yes, members can have their own blog – it’s a standard feature of a WordPress Multisite install.

Follow the instructions here to set up Multisite: https://codex.wordpress.org/Create_A_Network

Then check everything works, including custom permalinks.

Install BuddyPress (automatic plugin install)

And, hey, you will have an admin blog, and separate blogs enabled for members