Yes, that is correct. That is one of the constraints of stateless share-nothing processes. This is also described in one of the principles in "The Twelve-Factor App": https://12factor.net/processes. All common/permanent state must be stored in common storage for all the processes.
Depending on the kind of permanent storage needed, you could use one or more of the managed databases offered by Google Cloud. Here is an overview: https://cloud.google.com/products/databases/. You could also provision another open source database yourself either in Compute Engine or Kubernetes Engine, e.g. MongoDB. There is a lot of these on Marketplace, again for example MongoDB: https://console.cloud.google.com/marketplace/details/click-t....
In an effort to share knowledge by writing daily articles in our 12 different advent calendars at https://bekk.christmas I contributed with this article in https://thecloud.christmas. It's just a short introduction to Cloud Run, how it compares to alternatives such as FaaS and Kubernetes, and some info on how to build and deploy to Cloud Run. If you want to learn more about Cloud Run there are several links to both the Cloud Run announcement blog posts from Google and the official documentation. Hope it can inspire someone to check it out for their serverless journey. I have been using it for a while for smaller services that integrate with both Cloud Datastore and Cloud Pub/Sub, and I think it works really well :)
Depending on the kind of permanent storage needed, you could use one or more of the managed databases offered by Google Cloud. Here is an overview: https://cloud.google.com/products/databases/. You could also provision another open source database yourself either in Compute Engine or Kubernetes Engine, e.g. MongoDB. There is a lot of these on Marketplace, again for example MongoDB: https://console.cloud.google.com/marketplace/details/click-t....