When the remote MySQL host has no space to back up its database, the backup file can be stored in another box that has enough space.
Log in to the box with enough space
Run mysqldump to copy the records from the remote MySQL host to the current box.
mysqldump 'target_host_ip' -u'target_host_mysql_user_id' -p'target_host_mysql_passwd' > backup_file_of_target_host.sql
The backup MySQL file (backup_file_of_target_host.sql) is created on the current box.