6.83 Exabytes

According to my calculations, the average adult human body stores about 6.83 EB (exabytes) or 6 830 PB (petabytes) or 6 830 000 TB (terabytes), which is equivalent to around 427000 16-TB hard drives.

Calculation Part: (If you want to see the process...)

We can interpret "fraction of an ounce of DNA" as "fraction of an ounce of base pairs" since DNA is composed of base pairs. Let that be N ounces of base pairs = 0.3 EB (exabytes) or 300 PB (petabytes) or 300 000 TB (terabytes) data. Now let's incorporate base-pair data from the human genome (https://en.wikipedia.org/wiki/Human_genome). There are around 5,976,710,698 base pairs (considering that we are diploid) in ONE SOMATIC cell. According to the Nobel Site, there are about one hundred trillion (1 x 10E+14) cells in the human body. The average weight of a base pair is about 650 daltons. Therefore the average number of grams of base pairs the average adult human has in his or her body is:

1 x 10E+14 cells * 5,976,710,698 base-pairs/cell * 650 daltons/base-pair * 1.6605389 x 10E-24 grams/dalton = 645.0964395 grams of base pair
Since 1 ounce = 28.3495 grams, N ounces of base pairs = N * (28.3495 grams of base pairs).
N * (28.3495 grams of base pairs) = 0.3 EB (exabytes)

Let X be the total amount of data an average adult human body can store:
X = 334.1338181 grams of base pair * [0.3 EB / (N * 28.3495 grams of base pairs)] = 22.75512582/N * 0.3 EB = 6.826537747/N EB.

If we approximate N to 1, then X is about 6.83 EB. There is some percent of error since I assumed all cells to be somatic.
http://www.sciencealert.com/here-s-why-dna-could-eventually-replace-hard-drives Here's why DNA could eventually replace hard drives