Great question. This is one of the questions that my friends and I have asked at various points in our lives. (also: how long would it take for N monkey, typing randomly, to create Macbeth spontaneously?)
The population data SEEMS wrong. Why? My suspicious arise because there is no mention of estimated lifespans or the rate of death in ancient populations. Let's try to fill in some of these and see where that gets us:
Lifespan: let's estimate 40 years. (By this measure, Jesus was normal by actuarial standards)
Death rate: if the lifespan is 40 years, then for every cohort of 1000 people, there would be 25 deaths per year (1000/40 = 25).
Plug these numbers into a spreadsheet and see that a starting population of 5M in 8000 BC would grow to 5e192 by the year 1 AD. !!! SOMETHINGS WRONG HERE !!!
Looking more closely, at a birth rate of 80 per 1000 per year (and a death rate of 25 per 1000 per year), 5M people would grow to 10M in 13 years. This is much higher than the flat line shown in all of these growth models.
If you assume the rate of birth is steady at 80/1000, you need to jigger the death rate to over 79.48 to achieve the stated population of 300M by 1AD.
I'm thinking that the number of births is too high: one would have to look at the model demographics to see if this is realistic: Assume an evenly-distributed cohort of 1000 persons, with ages ranging from 0 to 40. this places 25 people per age. Assume that the fertile years are from 15-40, etc. and you can really look at a more realistic model of populations. (for example, half of those born would not be able to give birth...)
To sustain a lot of births and keep a slowly growing population, you need to have a lot of deaths. We need numbers like infant death rates and the risk of death during childbirth. If they are huge (79.48 / 1000), this can get you to 300 million by 1 AD on a smooth glide path. Doing so would require 47 billion people to be born in this span--a truly epic slaughter. (this might be the way that the referenced numbers were achieved)
It is possible, though, to look at the rate of birth and death in primitive populations in the Amazon, New Guinea, or Kalahari tribesmen. I suspect that this would suggest birth rates at 50 per 1000 (I'm not asserting this--i'm probably wrong); It's probably not a good idea to gauge ancient birth rates from any data collected anywhere after 1800. Increasing urbanization, medical, food improvements, and access to petroleum-based energy added an artificial stimulus to growth rates that probably did not exist in earlier eras.
With these numbers, it would only require 28 billion people to be born in this time.
My suggestions are: (1) define a model that accounts for realistic birth behavior and more detailed methods of death. See what models suggest from this. (2) try to find some proxy for historical birth rates form archaeology or comparative anthropology.