Yes, ATGC are the 4 bits, but in order to create something useful, say Lysine, you need AAA (three of them, forming a codon).
With computers, a 3.3 volt, a 2.9 volt are 1 and 0 volt, 1.6 volt are 0 and these are grouped together usually by 8, with 2 of these, "00 41" we have the letter "A" in Unicode. But we need a single one of those 8 to get an ASCII code.
So a "bit" in DNA is Adenine, and a "bit" in computers is just a given amount of voltage, with a high refresh rate which represents zero or one (or true/false).
We cal these
Quaternary bits and Binary bits, respectively...
but it's still a SINGLE bit (even if Quaternary), but to represent a Quaternary bit in Binary, you will indeed need more binary bits.v
So: 4 combinations grouped_by 3 = letter and 2 combinations grouped_by 8 is letter.
>I'm just not sure if pairs beginning with adenine or whatever can be repeated in sequence,
Yes they can. Any order is possible. They've even encoded
video in DNA. See the CRISPR method on DDG for more details.