In the first century, a person could choose to become a physician (doctor), just like we can today. But back then, people did not understand nearly as much about the body as we do. Most physicians in the Roman Empire were brought there from Greece, where they were often from noble families and were highly respected. In the Roman Empire, however, they were not always respected as much, because they were immigrants and usually either slaves or freed slaves. Also, in Bible times, unlike today, no one really checked doctors to make sure they were treating people correctly instead of hurting them more. Many did more harm than good (Mark 5:26), and some doctors even mixed in magic and spells, as though that would help. Add to that the fact that physicians were usually thought to be greedy in charging their patients unfairly high prices to help them, and you understand why they were not always respected.
In spite of that, some physicians were good at healing and helping. Luke, who wrote Luke and Acts, was a physician (Colossians 4:14). In comparison to other inspired writings, his writings are filled with extra information about sicknesses and physical conditions that showed he had studied medicine. Jesus mentioned doctors in a favorable way (Luke 2:17), and in fact, is known to us as the Great Physician (Mark 2:17) Who healed many people that no other physician could ever help.