QuiteRSS. It's very self hosted in the sense that it's a normal native Qt GUI application for running on Windows, most Linux distros, MacOS, FreeBSD, and OS/2. No need for fragile complicated stuff like a webserver, dynamic language execution, database server, or extra programs required to use it (ie, browsers). It uses sqlite internally and sqlite tools work with it if you want an SQL interface. It handles my 1000+ feeds pretty well.

It has no consideration for mobile support or concepts like synchronizing.