Just upvoted Robin Craig's answer to Can two people repopulate Earth? Is it possible?  Makes it clear that it is possible. Thought I'd fill it out a little more with some images and hopefully draw attention to it as it seems there are many answers here being upvoted by people who are not aware of this study and say it is impossible:

The second paper he gives is particularly clear. A pair of mouflon (wild sheep)

One Mouflon Ram

Ovis orientalis - by Jörg Hempel

Mouflon Ram

Mouflon ewe  Ovis musimon by Doronenko

Not those actual individuals - they are for illustration purposes. But just two wild sheep introduced to an island formed an entire population which was not only healthy but increased in genetic diversity. Unexpected heterozygosity in an island mouflon population founded by a single pair of individuals

Humans have the advantage as others have said that they would know that they are an endangered species and would decide what to do about it.

Obviously would have to have a lot of inbreeding. It's possible some humans might decide this is not acceptable and they'd prefer to be extinct. But biologically it seems feasible.

SCENARIO?

The question doesn't give a scenario. Hard to think of something realistic that would kill all humans except for one couple. Or that would lead to everyone dying a natural death except one young couple.

Even an illness that sweeps the world would leave some survivors, including for instance uncontacted indigenous people. Things like artificial life escaping from the laboratories and reproducing in the wild and replacing Earth DNA with XNA based life throughout the world would kill everyone and would still be there at the start of the plot.

I see it as science fiction rather than future possibility. So then in those science fiction scenarios where only two people are left alive, then sometimes they make it so that everyone dies except them in some plague that affects the entire world. They are, say, doing an experiment in living underground for years without any contact with anyone else, including no communication - then when the experiment is over, they come to the surface and find they are the only people left in an abandoned world. That sort of thing. So they have the technology, have books to recover their knowledge etc, but only the two of them.

Or. you can suppose that they are genuis scientists that find the solution to reversing the thing that kills absolutely everyone in the world, but too late to save everyone except themselves.

Another possibility from science fiction is the idea of an aged population where they have artificial ways of giving birth, and most people typically live to some great age like a million, but new children are born, say just two of them. The rest of their society dies of old age. These are the only ones left. And they make a fresh start.

It's one of the most overused tropes in pulp fiction. Adam and Eve Plot - TV Tropes .

Sometimes they are in a world with no technology and many dangers, in which case they probably have little chance, but sometimes they are surrounded by high technology, automated hospitals, libraries. Or maybe are in a starship with all the technology of the starship at their disposal. I'm assuming a scenario like that.

And - a bit more clarification got from a discussion here: How many people are required to maintain genetic diversity?

IDEA OF A MINIMUM POPULATION AND PROBABLE NEED FOR PLANNED BREEDING

The idea of a minimum population is based on the idea of unplanned breeding, where anyone mates randomly with anyone else of the opposite sex, and is a probabilistic thing.

If you can plan who mates with who, then - perhaps one couple has enough genetic diversity, if unrelated originally and they don't have some serious defect for their children to inherit.

They would be able to build up to a population of a few hundred, and then would be able to survive after that, except, that the entire population would be vulnerable to some virus or other condition they are not adapted to until they have survived long enough for some adaptations to arise through actual genetic mutation rather than just shuffling existing genes.

And the example of the sheep would seem to show practically that if you are lucky, then two would be enough if you are lucky with the individuals.

For another example of a species that has survived a major bottleneck, nearly all golden hamsters

Peach the pet hamster

originate from a single litter. Golden hamster

I know it depends on the species as well. You can't draw conclusions from humans directly from sheep or golden hamsters. But these are a few points that seem to have been left out of the top rated answers here, maybe someone with expertise can say more about it?