Is it possible to backup a database that is not ONLINE?
It appears that no you can not backup a database in an offline state based on this comment from BOL.  Offline and Emergency appear to be the exceptions.

Without the NO_TRUNCATE option, the database must be in the ONLINE state. If the database is in the SUSPENDED state, you might be able to create a backup by specifying NO_TRUNCATE. But if the database is in the OFFLINE or EMERGENCY state, BACKUP is not allowed even with NO_TRUNCATE. For information about database states, see Database States.

However that being said the chances of the database state being the reason for your backup problem are slim.  As said in the comments by @StrayCatDBA MSDB is required to do backups (among other things).  You can look at the error for a backup by right clicking on your maintenance plan (assuming you are using one) and selecting view history.



Once you are in the Log File Viewer you can see what errors you have.  Click on the + to drill down to details and then select each of the lines with the red X in turn to see what the current error is.



In this particular case we had a maintenance plan that was trying to back up databases that no longer existed.  Once they were removed from the plan we stopped getting the errors.