Not OP but I can setup DB of choice on an instance or group of instances and access directly from the Cloud Run containers. Our particular main stack is on MongoDB 3.4 so we deployed Atlas on the VPC and the containers access it just fine.

I use Cloud Firestore, it's also serverless and is really easy to use for building SPAs as you can do a lot of things with just frontend code and trigger cloud functions when things in the database change.

I do wish there was a serverless SQL option like AWS has, but nothing yet. Can always use Cloud SQL, but as others have mentioned you need to access it over a public IP (for now)