In recent years, there has been growing popularity of applying drones for medical services. Such applications always involve flying safety, data transmission, and personal privacy; therefore, reliable communication and enhanced information security should be addressed first. The 5G cellular network and blockchain technology emerge as promising candidates for solving these problems. This article first categorizes the medical applications of drones, and then presents some key challenges in these applications. Then we give preliminaries of 5G and blockchain, and explore their intrinsic characteristics to provide reliable communication and enhanced information security. In the scenario of disaster relief, an illustrative example is presented and discussed. Analytical results demonstrate the great potential of using 5G and blockchain for promoting the medical applications of drones.