Exchange 2003 server failing, writing 2 gb of log files per hour

We are in serious crisis mode.  We have an Exchange 2003 server.  Because of lack of controls on it during the previous staff's tenure, it has grown to 340 Gb.  We are moving it to Exchange Online in a couple of months, and are going to be pruning it down.  However, yesterday it started dismounting the stores.  They'll restart, but only for a couple of hours.  We then noticed that the backups were failing.  We now can't run a backup, and the log files are going crazy.  We have ran out of room, and have moved some of the files (I know you shouldn't do that).  But they are not getting flushed because the backup is failing.  All of a sudden, we're getting 2 or 3 Gb of log files an hour.  

The backup report reads The 'Microsoft Information Store' returned 'Error returned from an ESE function call (d).
' from a call to 'HrESEBackupRead()' additional data '-'

When the backup stops, we get the error listed below.  

Event Type:      Error
Event Source:      ESE BACKUP
Event Category:      Callback
Event ID:      904
Date:            7/5/2013
Time:            3:19:36 AM
User:            N/A
Computer:      TOH
Description:
Information Store (3344) Callback function call ErrESECBDoneWithInstanceForBackup ended with error 0x8004010F .

Any body have any ideas?  I need to keep this database going for another month or so.  But even our backups are in question due to the size of the database.