The first question you're going to ask is "If you can fit 700 TB of data into a gram of DNA, and the human body has 6-60g of DNA in it, how much data is encoded in human DNA?" According to Wikipedia, the source of all truth:
The haploid human genome (23 chromosomes) is estimated to be about 3.2 billion base pairs long and to contain 20,000–25,000 distinct genes. Since every base pair can be coded by 2 bits, this is about 800 megabytes of data. Since individual genomes vary by less than 1% from each other, the variations of a given human's genome from a common reference can be losslessly compressed to roughly 4 megabytes.
There are two numbers here:
800 MB is approximately the upper bound on amount of information that 3.2 billion arbitrary base pairs of DNA can hold.
4 MB is an upper bound on the true "information content" of a given individual human's genome, given that we know it's a human genome.
To explain that, we need to back up to what megabytes and terabytes really are. A "bit" is the basic unit in which information is measured. 1 bit = the amount of information needed to describe the result of a fair coin flip (i.e., either "heads" or "tails.") 2 bits is the amount of information needed to describe two coin flips ("two heads", "heads then tails", "tails then heads", "two tails"). Actually, describing anything that has four equally probably outcomes is two bits worth of information - so telling someone whether a DNA base pair is adenosine, thymine, guanine, or cytosine is still two bits of information. That's what Wikipedia means when it says "every base pair can be coded by 2 bits." And from there, 8 bits = 1 byte, 1024 bytes = 1 kilobyte (kB), 1024 kB = 1 megabyte (MB), 1024 MB = 1 GB, and 1024 GB = 1 TB and so on.
The key here is the information needed
to describe your coin flips or what have you. I could write a thousand page novel about the result of a coin toss, or I could grunt once for heads and not grunt for tails - it's still one bit of information either way. Just as mailing one pound of merchandise with ten pounds of packing peanuts doesn't give you eleven pounds of goods, being inefficient in communication/storage doesn't mean you are handling more information.
There are two reasons why the total informational content of a given human's DNA is so much lower than you'd predict by multiplying 700 TB/g * grams DNA. The first is that DNA is really, really redundant. Every cell* in your body has a copy of the same 3.2 billion base pair DNA. Remember, information content is only the absolute minimum needed to describe something. Since every cell has the same DNA, you can describe the whole body's DNA by describing one cell's DNA and then saying "...and then repeat that again for every cell in the body." That message doesn't get any longer or more complicated the more cells you add, so the information content of a whole human body's worth of DNA is the same as the information in a single cell. 1 cell has 3.2 billion base pairs which is at most 800 MB of information.
The second reason is that natural DNA is very predictable. The base pairs aren't random sequences; there are a lot of cases where if you were to read a sequence of base pairs to a biologist, she'd be able to guess with very high accuracy what the next set of base pairs are. For example, if you had the first half of the ALB gene, you'd know that the next set would have to be the rest of the ALB gene. Further, she'd know that we were discussing a human genome, since ALB is the human version of a gene found in all mammals, and therefore she'd know all of the other genes that are common to all humans were present, etc. Just saying "This DNA came from a real human" would tell here 99.5% of what she'd need to know to reconstruct that DNA.** All she needs after that is the ~4 MB of information which describes how this particular human's DNA differs from the theoretical "average human" DNA. Since we only need 4 MB of information to fully describe a human's DNA to the appropriate expert, the actual
information content of the DNA in a human body is at most 4 MB.
*except sperm and eggs, which only have subsets of a normal cell's DNA. And damaged / mutated cells, of course.
**Humans, weirdly, have a lot less genetic diversity than other mammals. There's a theory that humans went very nearly extinct - down to thousands of individuals - before leaving Africa, and that maybe only a thousand people actually left Africa in the initial colonization of Eurasia. Which makes us all inbred as fuck.