For Databases it depends on the Database/Use case:
Postgres: Postgres BDR with HAProxy
MongoDB: Replica Sets and modify API URI to mongodb://db1.example.net:27017,db2.example.net:2500/?replicaSet=test
Though Mongodb can be used with HAProxy, the Mongodb's internal failover system has to be disabled and it would probably be easier to just change the URI as stated above.
The native replication/failover functionality needs to be used for the database and sometimes proxied to.
For other persistent Docker volumes, DRBD would probably be sufficient.