That error is coming from the SSIS runtime, not SQL Server.
Running out of space in SQL Server produces
Msg 9002, Level 17, State 4, Line 20
The transaction log for database 'XXX' is full due to 'ACTIVE_TRANSACTION'.
or
Msg 1105, Level 17, State 2, Line 20
Could not allocate space for object 'YYY' in database 'XXX' because the 'PRIMARY' filegroup is full. Create disk space by deleting unneeded files, dropping objects in the filegroup, adding additional files to the filegroup, or setting autogrowth on for existing files in the filegroup.
It's unrelated to storage, and usually indicates a memory problem. I would first try reducing the buffer sizes in your Data Flow, and ensure that your data flow doesn't have any components that require loading large amounts of data into memory, like lookups.
See Data Flow Performance Features