Thanos doesn't have production support for local storage either. The only stable storage providers for it are google, amazon, and azure's object stores.
Interestingly, it looks like Cortex's support for local storage and object stores comes from using Thanos's storage engine. So once it's production ready in Thanos it will probably be production-ready in Cortex shortly thereafter.
https://cortexmetrics.io/docs/operations/blocks-storage/
I think for Cortex your safest storage options now are Bigtable, DynamoDB, or Cassandra.