Data

Estimated number of eukaryotic (human) cells in the human body: 1.0×1014
—Wolfram Alpha

The haploid human genome (23 chromosomes) is estimated to be about 3.2 billion base pairs long.
—Human Genome – Wikipedia

The full DNA content of a cell is therefore 6.4 Gbp.

Average base pair size: one bp corresponds to approximately 3.4 Å of length along the strand
—Base Pair – Wikipedia

Average Pluto distance from Earth: 6.09×1012 meters
—Wolfram Alpha

Average Pluto distance from Sun: 6.09×1012 meters
—Wolfram Alpha

Note: Why are these basically the same distance? Think about it: 50% of Earth's orbit is behind the Sun so it adds to the distance, but the other 50% the Earth is in front of the Sun so it subtracts. On average, it's a very similar distance.

A Pluto round trip is approximately: 1.2×1013 meters

Average Moon distance from Earth: 3.85×108 meters
—Wolfram Alpha

A Moon round trip is approximately: 7.7×108 meters

Calculation

Total length of DNA in 1 cell:

6.4×109 × 3.4×10-10 meters ≈ 2 meters

Total length of DNA of all cells in human body:

2 meters × 1014 = 2×1014 meters

The total length of the DNA strands in the human body in units of Pluto roundtrips is:

2×1014 meters / 1.2×1013 meters/Pluto roundtrip ≈ 17 Pluto roundtrips

10 billion miles are 1.6×1013 meters (assuming statute miles for "miles" – you never know exactly with uncommon units)

The total length of the DNA strands in the human body in units of 1x1010 statute miles is:

2×1014 meters / 1.6×1013 meters/1010 statute miles ≈ 12×1010 statute miles

The total length of the DNA strands in the human body in units of Moon roundtrips is:

2×1014 meters / 7.7×108 meters/Moon roundtrip ≈ 2.6x105 Moon roundtrips