This app was mentioned in 3 comments, with an average of 2.00 upvotes
I'm google-free for the most part and use my paid apps even without Play. It's a bit of a hassle but easy once set up: ApkTrack lets you know when Play apps are updated, download the APK with Google Play Downloader and sync the APKs with a NAS. You'll need a good wifi sync app like DS Cloud or DAEMON Sync.
For calendar/contacts use DAVdroid from f-droid.
Unfortunately you'll need workarounds for any purchased Play apps, LP works but I'm open to better suggestions.
Synology supports this natively with the DS Cloud Android and iOS applications. Just need to have the Cloud Station service installed on the NAS.
Use DS Cloud instead, it does for me what you're asking