It's probably just an estimate, but is reasonable given other physiological data available.
According to this chart, a 50th percentile, middle age man is about 180lb (80kg). From Harvard's bio numbers, the weight of a skeleton, is typically about 15% of a person's weight. This leaves us with about 150lb (70kg) of tissue to be filled with blood vessels.
This well referenced article shows that the tissues remaining all have densities of approximately 1g/cm^3.
If you take the 60,000mi value at face value, you could calculate the upper limit (because your mass is not 100% blood vessels) for the average diameter of a blood vessel by the following equation:
(1 g/cm^3) (100,000 km) (pi/4 d^2) = (70 kg)
d = sqrt( (70kg) / ((100,000 km) (pi/4) (1 g/cm^3 )) )
d = 30 microns
According to the University of Minnesota's page on vascular physiology, capillaries are the most numerous, make up much of the length of the vascular system, and average 5-10 microns in diameter.
Since the plausible upper limit on average blood vessel diameter is 30 microns, it seems reasonable that there could 60,000 miles of blood vessels in the human body when capillaries will try to pull that average toward 5-10 microns.
I'd also like to point out the wide range you're likely to see in real life. The University of Minnesota page, for instance, figures there's about 25,000 miles of capillaries. I suspect there are large error bars but that the estimate is roughly correct.