Currently, the main challenges of the frameworks for healthcare monitoring are as follows: minimising latency, especially for delay-sensitive diseases such as sudden heart disease; identifying health situation in a timely and accurate manner when correlating physiological indicators and context information; reducing the risk of exposure because health data are highly private. In response to the above, this paper proposes a framework for 5G-secure-smart healthcare monitoring (5GSS) to achieve the following goals: fast and accurate identification of context-aware health situation, blockchain-based secure data sharing, and low-latency services for emergent patients. The framework consists of a data acquisition layer, a diagnosis and security layer (edge cloud), and a health service layer. The proposed framework adopts the following key technologies: a 5G-IPv6 communication network, context-aware health situation identification-based similarity measure, and blockchain-based secure data sharing mechanism. Finally, a prototype system has been implemented to monitor hypertensive heart disease, confirming its effectiveness with respect to a real scenario. Combined with the data of 45 patients, the prototype system can identify health situations with an accuracy of 96.34% at a sensitivity of 92.46% and a specificity of 93.62%, while significantly reducing the latency and improving the data sharing security.