barman + postgresql not start after barman retore
It looks to me like you are trying to restore to a point in time that is earlier than what is recorded in the backups. 

Make sure you have not specified a recovery-time that is earlier than the backup you are trying to restore. 

Check the files recovery.conf and backup.label in the postgres data directory.
If you want to restore to the latest possible, then you should make sure you don't have a property named recovery_target_time in the recovery.conf.