IMHO Barman always uses a replication slot, so there must be an active
Barman slot on your DB server with xlogs that Barman has not yet received.
If the DB is a virtual machine, maybe you can increase a bit the storage
volume where /var/postgres/9.5/main/baran_xlog is located, and then look
at Barman as to why it has stopped receiving the xlogs, maybe it's full,
too, or there's a network issue.
If you increase the DB storage just a bit, it will make it possible to
start Postgres, but you need to solve the Barman problem very soon, or
else the space on the DB server will be filled again. If you can afford
to loose backups, you can also drop the slot.