Barman appears to be a solution for backups, not HA. If your master fails, you need to restore from the backup, which means downtime.
Barman can keep a hot-standby for you.
You still need a mechanism to swap over to the standby on failure. However with libpq from postgres 10 you can provide multiple servers. This isn't true HA, but for most people it fits the requirements.
From their FAQ (http://www.pgbarman.org/faq/):
Does barman manage replication and high availability as well? How does it compare with repmgr, OmniPITR, walmgr and similar tools?
No. barman aims to be a pure disaster recovery solution. It is responsible for the sole backup of a cluster of PostgreSQL servers. If high availability is what you are looking for, we encourage you to use repmgr. barman specifically targets the DR case only