There’s a similar topic with different alternatives:
About your specific problem with Google Calendar app, it’s normal. (Most) Google apps require a piece on Google Play Services to sign-in to a Google account. Without that piece (i.e. without the whole Google Services bloatware/spyware/backdoor), you can’t login.
Fortunately, the sign-in piece is implemented in microG, a free, libre and open source (FLOSS) implementation of Google Mobile Services. It worth a try, I guess,
