Google Cloud Run is a managed serverless platform from Google Cloud on which you run containers and source code without managing servers. Each Cloud Run resource runs in a region of your choice. The service is available in many regions, including several European locations such as Finland, Stockholm, Madrid, Belgium, Frankfurt, the Netherlands, Milan, Paris, Berlin, Turin and Warsaw, as well as Zurich, London and regions in Asia, the Americas, Africa, the Middle East and Australia. It is offered only as a cloud service.
Cloud Run is a Google product, part of Alphabet, headquartered in the United States. The service is therefore subject to US law, including the CLOUD Act, even when you choose a European region. Google offers a Cloud Data Processing Addendum in which Google acts as processor and which refers to the EU and UK GDPR. An EU region determines where data is stored, but not who has legal control over it; European organisations with strict sovereignty requirements should weigh that.