With lots of high tech & genetic engineering that the OP has ruled out.
And using quantities of Handwavium that would make most Star Trek & Star Wars writers blush. IOW, it was piled high and deep from corner to corner.
It was the OP that referenced Adam and Eve. While some scriptures mention Adam and Eve none of those books AFAIK are taken as fact in the sense you mention, however it is what the OP used as a standard to compare. Thus fair to mention the context of one of the scriptures (the bible, and btw not too far off from the Quran). On that basis I made my comment and thus I feel is valid to mention in this forum.
Without genetic tech, they’re going to have too many problems from inbreeding. According to people doing calculations based on genetic drift , you need somewhere from 80-160 people to start your repopulation to be safe. That’s about the size of a small village, which is also large enough to function reasonably well without ‘hunger, thirst, shelter, and protection’ provided as long as they have training in primitive survival or can learn it before the protection runs out.
The fictional story of Noah’s ark was written long before modern understanding of genetics and isn’t relevant here.
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I have heard that the kibbutzim dealt with this by have two creches, and keeping the children in each creche mostly apart from the children in the other creche.
Tell me this would not be going somewhere with his sisters, that’s the rhyme that first popped into my head 
They also had access to genetic engineering, which helped quite a bit.
He looked around to find a mate
But all the girls were his sisters
I find it hard to believe that the incest aversion would be strong enough to combat sex drive.
It’s one thing if, in a cultural and societal context where other mates are available, incest aversion causes you to choose them over possible mates you grew up with. But in the hypothetical, there aren’t any other mates to choose from.
I expect that the loss of genetic diversity resulting from a population bottleneck of just 2 people would likely doom them before long, even ignoring things like bad luck/accidental death or infertility causing a population dead-end very early on.
The thing about not finding people one grows up with to be sexually attractive/possible mate is called the Westermarck Effect, just in case anyone want to google it.
I’m sure there are plenty of cases of a small number of people who “go forth and multiply” and create a large population. The example that comes to mind is Easter island, where most likely one canoe-load of people arrived about 800AD or so. The path to there comes from the Pitcairn group, which is at the far end of the full Polynesian expansion; that doesn’t bode well for genetic diversity. Yet there are no suggestions of genetic anomalies.
Easter Island became a large community with estimates of up to 15,000 population before it crashed. Probably this was due to environmental factors and overpopulation. Various theories abound.
The same must be true of numerous other populations - how many people made it to the Andamans, Australia, or even North America in prehistory? How many from the North American genetic pool jumped the swamp in Panama to make it to South America? Tierra del Fuego? There’s Iceland from Scandinavia, then Greenland from Iceland…
To my mind, the real question is - how clean is the genetic material of the starter set? If there’s a propensity for diabetes, or breast cancer, or heart defects, etc. etc. - then all bets are off. Presumably if we’re talking passengers to a new world in a giant spaceship, they were selected for genetic quality. (Cf. Cyril Kornbluth’s “Marching Morons”) Plus, the Westermarck Effect says something about preference. Presented with very familiar matches and people who are new and different, it seems the latter appear more attractive. Presumably sibling rivalry issues also factor in… Quite a few medieval villages, people mixed and matched with fellow villagers, although I understand that trading women was not an unusual activity for many nomadic tribes (as was simply taking them). Plus, plenty of court cases demonstrate that sexual urges can overcome social familial restraints.
Not that genetic problems show up much. I recall the news articles about Fritzl, who kept his daughter locked in a cellar and had 7 children with her. One died not long after birth. He was discovered when one at 19 started having seizures, but there’s no indication that it was inherited genetic in nature. As for birth, women were doing that long before hospitals. It probably helps if there’s someone who knows something about childbirth, like umbilical cord issues and basic sanitation. Humans, thanks to bipedalism and hip arrangements (and their big fat heads!) are the only species where the mother often needs assistance to give birth, although there are plenty of cases of women giving birth alone. the other issue is that breastfeeding can help suppress the chances of future pregnancy, so weaning early might be another tactic necessary.
So for the OP - if it’s a pair of decently health adults, and they go at it early and often, they probably can do a decent job of repopulation; but until the population reaches a critical mass (let’s say, about 20 to 50) some problem could easily derail the attempt. The question is - are they deliberately trying to do so - working on a new pregnancy every year? Do they know about the right timing in ovulation cycles, etc.? Or just doing what they do when they get around to it? There’s a big difference between “they had 4 children and then…” versus “they had 12 children, and then…”
Should also point out that having many children ensures that much more of the original genetic material makes it into subsequent generations.
Remember that every human alive today can trace their ancestry back to a single female often called Mitochondrial Eve. For the male there is the Y-chromosomal Adam.
Almost certainly this Adam and Eve never knew each other but the point is we all come from one mom and one dad.
I really wish I could find an article I read that said if you go far enough into the future you will either be the ancestor of everyone or no one. Most likely no one.
Most likely everyone, actually. If you’re the ancestor to no one, it’s almost always because you didn’t have any children.
I think you missed what was being said.
Were you saying that if you go far enough into the future, it’ll be no one, because eventually humanity will go extinct? Because other than that, I can’t seen any interpretation that leads to “you’ll most likely be the ancestor of no one”.
Also consider the whooping crane - they were allegedly down to about 15 birds, and since have rebounded to near 1000, despite having serious problems succesfully raising offspring 9and still occasionally poached).
The cheetah gets its genetic non-diversity from apparently 2 population crashes, one about 100,000 years ago and one 12,000 years ago. I haven’t found any discussion of how many in the small group; one article says there are two ranges - South Africa and West Africa, and South Africa has the smallest diversity.
Humans also apparently went through a population bottleneck about 70,000 years ago, due to climate change from a giant volcano in Sumatra - although some argue i was a “long bottleneck” over 100,000 years. The theory is that population declined to between 2,000 and 10,000 humans at one time. Still, this is far more than a single breeding pair, but given the diversity of modern humans suggests a pretty good rate of genetic drift.
Wikipedia says this:
A 2005 study from Rutgers University theorized that the pre-1492 Native Americans are the descendants of only 70 individuals who crossed the [land bridge between Asia and North America
This sort of makes sense - if there’s a narrow land bridge between two land masses, the first group to occupy that bridge are somewhat a block for further migrants, since migration is usually about moving to open hunting grounds. There would be no point in wandering into an area of Alaska already being overhunted by current occupants, especially if it was a less fertile environment. Whereas the excess population from that group would find it advantageous to move on to green pastures further on.
Pitcairn Island was settled by 9 Bounty mutineers, 6 Polynesian men and 11 women (and a baby girl). Apparently a few of the white men were too busy drinking or killing each other to add to the gene pool, but the community was becoming overpopulated by the time they were found.
No.
If you do the math everyone in the future will be able to trace their ancestry back to a single female and a single male (who almost certainly never met).
So, unless you are that one person everyone can point to as their common ancestor then there will be a time when no one alive is a descendent of you (assuming you had children).
I think you might be misunderstanding the concept of mito-Eve and Y-Adam. Mito-Eve is a common ancestor of all modern humans, but she’s not the only one. She’s the common ancestor through the purely female line: That is to say, if anyone takes their mother’s mother’s mother’s … mother, eventually they’ll all go back to that same woman. And she’s not even the only common ancestor through the female line (since her mother and her mother’s mother and her mother’s mother’s mother and so on would also all be a purely-female-line common ancestor); she’s just the most recent one.
Likewise for Y-Adam: If you take your father’s father’s father’s … father, eventually you’ll get back to him… and then to his father and so on.
But there are many, many other lines of ancestry. For instance, I’m almost certainly descended from Cicero, but the line might go through my father’s mother’s father (or at least, one of the lines: There are probably a great many).
How does this work? If the common ancestor “A” has two children, “B” and “C” and half of the descendants come from each one, then your statement doesn’t make sense.