
Natural DNA has 4 nucleotides ATGC at any position so rather than base 2 (binary) it encodes in Base 4 (Quaternary). Hachimoji DNA for example has 8 nucleotides ATGCPZBS so goes to base 8. This is without additional modifications. ASCII in binary each letter takes 8 electrons in base 8 it would take 3 nucleotides. Physically DNA is several orders of magnitude larger then electrons on a HDD but multiple possible outcomes at each position increase its storage density.
Then DNA is also able to pack itself together in 3D incredibly tightly. Each of our nuclei fits 3 billion nucleotides into a space just 6 microns across around 113 cubic microns. And that packing still allows for read write access.