The short answer is yes. People did take themselves when ill to places dedicated to the god of medicine, Asclepius. Called Asklepieia, these were usually temples with associated springs with mineral water thought to have healing properties. Wherever the cult of Asklepius spread so did these places of healing, from Thessaly to Rome, from Kos to Athens. Generally, only patients suffering from diseases thought to be "treatable" were permitted in these temples (such ailments could range from paralysis to baldness so it was a wide range of stuff); those who were undergoing childbirth or were about to die weren't allowed into the holy spaces of the temple. Once inside, they would undergo ritual purification, sacrifices, prayer, etc. as well as undergoing rest in a clean environment and more traditional medical procedures like bloodletting. There was also the occurrences of incubatio wherein the patients while sleeping have Asklepius, as himself or as a snake or dog, appear to them to heal them by touch or to offer instructions on how to heal.

The figure of Asklepius was not only associated with religious healing in temples. He was also thought of as the progenitor and protector a lineage of physicians who would treat their patients with more rational means. And, to this day, the American medical profession uses the snake and staff symbol of Asklepius as a symbol.

Sources:

-Compton, Michael T. "The union of religion and health in ancient Asklepieia." Journal of Religion and Health 37.4 (1998): 301-312.

-Croon, J. H. "Hot springs and healing gods." Mnemosyne (1967): 225-246.