We present CoVR, a novel robotic interface providing strong kinesthetic feedback (100 N) in a room-scale VR arena. It consists of a physical column mounted on a 2D Cartesian ceiling robot (XY displacements) with the capacity of (1) resisting to body-scaled users' actions such as pushing or leaning; (2) acting on the users by pulling or transporting them as well as (3) carrying multiple potentially heavy objects (up to 80kg) that users can freely manipulate or make interact with each other. We describe its implementation and define a trajectory generation algorithm based on a novel user intention model to support non-deterministic scenarios, where the users are free to interact with any virtual object of interest with no regards to the scenarios' progress. A technical evaluation and a user study demonstrate the feasibility and usability of CoVR, as well as the relevance of whole-body interactions involving strong forces, such as being pulled through or transported.