I would highly recommend using a managed service like RDS or Google Cloud SQL. I feel that the days of the dedicated DBA have passed. With todays managed services you can focus on your application and infrastructure.
With that being said, running database workloads on Kubernetes is perfectly fine and in most cases acceptable.
What is really boils down to is the StorageClass's available to you. In most default installs your only (at least at first) option is to use a hostPath in your PersistentVolume's. This is really an anti-pattern of high-availability in the event your node goes down.
When I deploy MySQL on-prem I use a glusterfs backed volume and hit the ground running. The rest is just a matter of configuration.