How long will the AWS RDS outage take when enabling backups?

There is no outage to enable automated backup in RDS Service. When you modify instance and enable backup retention period there will be an I/O suspension typically for few seconds for RDS service to initiate snapshot backup of storage volumes used by RDS db engine.

That answer is not quite correct. For RDS engines other than Aurora, the documentation is accurate: when enabling a backup retention there is a brief outage while we restart the database. It is just the time it takes to reboot the database.