If possible, try upgrading to 6.5, as it resolves a bunch of issues since the 6.0 release. See here
for details. This list is not exhaustive as there were other fixes made in 6.1 to 6.4 too.
If you would like to continue using 6.0, we need to determine the cause of the error. When SQL Backup encounters a critical error, it generates a stack trace of events leading up to the error. This stack trace is then stored in a file named SQBCoreService_<instance name>_bugreport.txt, and is located in 'C:\\Documents and Settings\\All Users\\Application Data\\Red Gate\\SQL Backup\\Log\\' on Windows 2003 and older, and 'C:\\ProgramData\\Red Gate\\SQL Backup\\Log\\' on Windows Vista and newer.
Could you please check if that file exists on your system? If it does, could you please send it to us? If the file does not exist, it may be because the SQL Backup Agent service startup account does not have rights to create the file in that folder. You can test this by manually creating the stack trace file by running the following command from Query Analyzer/Management Studio when connected to the SQL Server instance:
EXEC master..sqbutility 9997
If the file is not created, try granting read/write permissions on the abovementioned folder to the SQL Backup Agent service startup account.