Android phone keeps rebooting, how can I save my data?
I'm sorry, but at this point your only hope is a factory reset.

The symptoms sound like the data for one of the system apps has become corrupted. If you had any way to stop the reboot cycle, you'd have a shot at saving the data. Since even safe mode doesn't give you that, there's not much you can do.

I will suggest, for the future, saving all of your contacts, messages, phone logs, et al., to external servers, such as Google Contacts. That will also help when you upgrade your phone. There are several apps in the Google Play store to save text messages, phone logs, photos, videos, and the like to a remote server.