MySQL will not start Please help, I really need to recover my databases.

service mysql start start: Job failed to start

mysqld --verbose 140401 20:16:11 [Warning] Can't create test file /var/lib/mysql/developer.lower-test
140401 20:16:11 [ERROR] mysqld: File '/var/log/mysql/mariadb-bin.~rec~' not found (Errcode: 122)
140401 20:16:11 [ERROR] MYSQL_BIN_LOG::open_purge_index_file failed to open register file.
140401 20:16:11 [ERROR] MYSQL_BIN_LOG::open_index_file failed to sync the index file. 140401 20:16:11 [ERROR] Aborting


This is what I've gathered so far.
In the Linux command prompt, I ran this:

sh-4.1# perror 122
OS error code 122:  Disk quota exceeded
MySQL error code 122: Internal (unspecified) error in handler
sh-4.1#


Evidently, the first or next binlog cannot be created

Check your diskspace by running

df -h /var/lib/mysql


Clear up the disk so binlogs can be made