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Yes. Lots of companies run things like ElasticSearch and Cassandra in Kubernetes. And even sharded RDBMS's are doable now thanks to projects like Vitess, which famously underpinned YouTube. Basically if there's a mature k8s operator for the DB in question, it's pretty straight forward. The "use containers/k8s for stateless stuff only" approach is very dated.

As far as I understand Zalando does this. They have their own PostgreSQL operator https://github.com/zalando/postgres-operator.

Ondat does that, a bunch of people are using them. And Akamai acquired them, probably to scale their model