The accelerating aging population has led to an increasing demand for domestic robotics to ease caregivers' burden. The integration of Internet of Things (IoT), robotics, and human-robot interaction (HRI) technologies is essential for home-care applications. Although the concept of the Internet of Robotic Things (IoRT) has been utilized in various fields, most existing IoRT frameworks lack ergonomic HRI interfaces and are limited to specific tasks. This paper presents an open-source unified Internet of Humans and Robotic Things (IoHRT) framework with cloud computing, which combines personalized HRI interfaces with intelligent robotics and IoT techniques. This proposed open-source framework demonstrates characteristics of high security, compatibility, and modularity, allowing unlimited user access. Two case studies were conducted to evaluate the proposed framework's functionalities, evaluating its effectiveness in home-care scenarios. Users' feedback was collected via questionnaires, which indicates the IoHRT framework's high potential for home-care applications.