This paper describes Cybi, an inexpensive smart companion mobile robot for elderly and disabled people. The software architecture that enables users to perform telepresence and teleoperation tasks with Cybi is based on combining Cloud Computing technologies and Fuzzy Logic. On one hand, the robot can be remotely teleoperated from a thin client by executing a specific module of a distributed software component that uses ROS, WebRTC and Google App Engine. On the other hand, Cybi is capable of understanding simple spoken orders that allow it to navigate through an unknown domestic indoor environment. In both cases, a fuzzy reactive navigation software component is used with the aim of making the teleoperation procedure as easy and safe as possible for both elders' caregivers and elderly users themselves. Finally, several real tests are also presented, demonstrating that Cybi could be considered as a suitable robotic companion, since the proposed methods enhance the performance of teleoperation and telepresence tasks.