Ancient yogis knew nothing of the modern yoga mat, and probably simply practiced on floor rugs. Having creative options and innovative ways to practice makes yoga without a mat an opportunity, not a compromise. When you don't have a mat and just a small space to practice, there's potential for a new relationship with gravity, the ground and different surfaces that meet the contact points of your skin. Practicing variations of common postures can enliven your practice and create new neural pathways that prevent you from simply going through the motions and shutting down the potential for new adaptability and responsiveness.
The following exercises simply make use of a chair and walls, surfaces you can find in even the smallest of rooms. Carry out these exercises either as a full sequence or as individual poses.