Yes, the DNA in all organisms gets damaged frequently, be it by radiation, chemical mutagens or copy errors. What keeps data integrity at acceptable levels in living organisms is the repair mechanisms all cells have. Keeping your data storage very clean, and shielded from UV will reduce the decay, but eliminating copy errors and cosmic radiation is near impossible.
With serious redundancy and storage for a limited time I could see this work in the future for some types of data storage needs, but the mutations are indeed are a severe liming factor.