Anbox doesn't come with Google Services as Google Services can't be installed on an unregistered device / firmware, but there are ways to achieve that in Custom ROMs or even emulators. If you want to use Android to its fullest then Google Play Services is a must.
What have I tried so far?
I've tried to install Google Play Services x86 apk, which successfully got installed and also appears in Settings -> Apps.
I've also tried to install Google Play Store which also got successfully installed and appears in the App Drawer of Anbox (also in GNOME), but it doesn't work as expected. For example, the YouTube app loads content but can't play videos and Google Play Store doesn't even load content, I think maybe because Google Play Services isn't installed as system app or maybe because there are some other Google apps that these things depend on which I'm missing.