It seems to be possible to synchronize Google Contacts (CardDAV) and Google Calendar (CalDAV) using DAVdroid (thanks).
Please note that Google servers don't seem to be very reliable and synchronization with Google is not officially supported by DAVdroid. Using DAVdroid with Google does not provide any additional privacy or other advantages in comparison to the default Google synchronization, but you can use it without having GApps and/or a Google account on your device.
Google Contacts and Google Calendar
Base URL: https://firstname.lastname@example.org/events (replace email@example.com by your Google account)
If you need another calendar than the main calendar, you have to find out the calendar ID (something like firstname.lastname@example.org) and use this base URL:
User name and password: your Google account
You have to allow "less secure apps to access your account" because Google only considers OAuth to be "secure", while DAVdroid uses Basic/Digest auth over HTTPS (which would allow app-specific passwords, too).