Originally, this was a Roman (or possibly even Hellenistic) public bath. St. Demetrios used to preach to his fellow Christians near the Ancient Agora. As punishment, one day he was dragged away from the Agora and down to the fountain, where he was speared to death in front of the public. It has been a site of Christian pilgrimmage ever since, with the first large temple constructed on it around 450.
After the 10th century, the site of the fountain (now underground) became associated with the myrrh which poured from his relics. In the middle photo, Pelagia is standing next to the fountain. Pipes from upstairs end here; the myrrh from the relics used to drain through the pipes down to the fountain. The faithful would then come to this fountain to take some of the myrrh. Now the relics are encased, and I hear it is opened up once a year on his feast day to retrieve the myrrh that has accumulated. One American reported being able to smell the myrrh from BLOCKS away when this happens.