The rapid evolution of technology allows the healthcare sector to adopt intelligent, context-aware, secure, and ubiquitous healthcare services. Together with the global trend of an aging population, it has become highly important to propose value-creating, yet cost-efficient digital solutions for healthcare systems. These solutions should provide effective means of healthcare services in both the hospital and home care scenarios. In this paper, we focused on the latter case, where the goal was to provide easy-to-use, reliable, and secure remote monitoring and aid for elderly persons at their home. We proposed a framework to integrate the capabilities of edge computing and blockchain technology to address some of the key requirements of smart remote healthcare systems, such as long operating times, low cost, resilience to network problems, security, and trust in highly dynamic network conditions. In order to assess the feasibility of our approach, we evaluated the performance of our framework in terms of latency, power consumption, network utilization, and computational load, compared to a scenario where no blockchain was used.