There actually was a real Adam and Eve, although they didn’t live for 900+ years like it says in the Bible, and we don’t know their names (or even if they had them!).
All modern humans like you and me can trace our maternal (mother’s) line back in an unbroken chain to a woman known as Mitochondrial Eve. She lived around 200,000 years ago, most likely in east Africa. Without having a sample of the ancestor’s DNA, it is impossible to work out the genetic makeup of an individual who lived a long time ago. However, scientists can analyse many descendent’s genomes and work their way back.
Mitochondria are small power packs or batteries found in every cell in the body. They provide energy for the cell and actually contain a small amount of DNA that is not found in the nucleus. This mitochondrial DNA comes exclusively from your mother, as the only mitochondria in an early embryo come from the egg, not the sperm. By examining this mitochondrial DNA for spontaneous mutations that happen about once in every 3500 years, it provides a clock that leads back to Mitochondrial Eve, which is incredibly cool!
Ancestral Adam is another cool story. The Y chromosome is responsible for turning fetuses into boys. Even rare babies that are born with multiple X chromosomes will still be male if they have a single Y chromosome. This Y chromosome can be traced in line back to Ancestral Adam. It is transferred from father to son. He is thought to have lived around 142, 000 years ago, again probably in Africa. The comparison of human Y chromosomes with chimp and ape Y chromosomes is still ongoing and so it is possible that the individual Ancestral Adam may have lived even longer ago.
However, it is not necessary that Ancestral Adam and Mitochondrial Eve ever met or lived at the same time. It is just important that their descendants managed to pass on the DNA to their children before dying!