Comment by dstaley

Cloud Run user here. It does not support running a database as a Cloud Run service; you'd need to connect to a Cloud SQL instance. This is one area I really hope PaaS providers improve on in the future. The closest thing at the moment is Aurora Serverless on AWS, especially with their neat HTTP connection system. That being said, cold starts are even worse since they're now the start up time for your application in addition to the start up time for Aurora Serverless.