Synchronize google apps with private server? Is there a way to synchronize google apps with my own server? Does Google offer something like that? Maybe for business use?
Or is there maybe an unofficial way of doing it?

I couldn't really find anything about this. All I could find out is that you can use your own domain when creating a google account, but I don't think that this affects the data sync.
Yes and no. It's not possible to use your own server, but Google does offer a hosted platform for businesses called Google Apps for Work. It gives you a Gmail interface (both the webapp and the Android app) and Google accounts in your own domain (which you can register through Google or independently), along with a private Google Drive for your domain and some other of their apps.

All the data are still stored on Google's servers, but because it's an enterprise service, you have service level agreements, non-disclosure and confidentiality.