The beauty of modern game systems (and many, many other applications) is that your saved games do not depend on the CD-ROM (which stands for 'Compact Disc - Read Only Memory') for storage. Read only means that the Xbox 360 only reads from the disc, and does not write to it. The memory is in the Xbox's hard drive, which means that your Destiny save data is dependent on the Xbox, not the Destiny game disc. Granted, it will probably be another $15 to buy another Destiny disc (available on Amazon), but you will not have to start all over from the beginning.
If you need more reassurance, please refer to this question, which explains you concern in greater detail: Skyrim disc damaged, if I buy another will I still be able to play my saved games?