Tapered optical fibers (TOFs) are used in many areas of physics and optical technologies ranging from coupling light into nanophotonic components to optical sensing and amplification to interfacing quantum emitters. Here, we study the fundamental torsional mechanical mode of the nanofiberwaist of a TOF using laser light. We find that this oscillator features a quality factor of up to 10 7 and a Qf product of 1 THz. We damp the thermal motion from room temperature to 28(7) mK by means of active feedback. Our results might enable new types of fiber-based sensors and lay the foundation for a novel hybrid quantum optomechanical platform.