As John Hanley mentions, Cloud Run doesn't support background processing. Depending on your requirements, you have a few options though.

Option 1

Use a separate service to run your background jobs, potentially using Cloud Pub/Sub service-to-service messaging subscriptions to broadcast when the job is ready. Your web service can return a handle/ID to the subscription, which the caller can use to listen for updates.

The background service itself can be run in its own Cloud Run container, if appropriate, or since you appear to be using NodeJS, you could package it as a Cloud Function.

Option 2

Move your container to a solution that supports background processing, like App Engine or Kubernetes Engine (GKE). These two have very different pricing models to Cloud Run however, and depending on your usage patterns could end up being significantly more expensive. (This Google blog post breaks down the differences between GKE & Cloud Run.)

GKE can handle a single-container setup without any need for you to learn or mess around with Kubernetes directly. But if your project architecture changes, or you need to do any configuration or troubleshooting, there could be a significant learning curve involved.

Option 3

Rewrite your code to use synchronous processing instead, maintaining the request until the job is complete. This would only be feasible if your jobs will definitely always complete before the timeout (60s?), and if your end use-case allows for it. This one is the simplest solution, but the most restrictive and prone to user-facing errors.