Contact option was not listed in Android's (v.6.0.1, Marshmallow) Settings / Accounts / Google sync menu.
The problem was that the Contacts Sync application had no access right to my contacts.
To fix this, I switched the Settings / Apps / Gear icon / App permissions / Contacts / Google Contacts Sync toggle.
After that it started to sync, but I got the same 'Sync is currently experiencing problems'.
It turned out that I exceeded the Google API call limit for the day, so after 24 hours it synced successfully.
To figure out what the problem exactly was I connected my phone through USB to my PC, enabled debugging on the phone, used the `adb logcat` command from the Android SDK and examined the errors/exceptions in that log. This way you can get to the bottom of your calendar issue as well.