Stimulated Brillouin scattering (SBS) has many applications; for example, in sensing, microwave photonics, and signal processing. Here, we report the first experimental study of SBS in chiral photonic crystal fiber (PCF), which displays optical activity and robustly maintains circular polarization states against external perturbations. As a result, circularly polarized pump light is cleanly backscattered into a Stokes signal with the orthogonal circular polarization state, as is required by angular momentum conservation. By comparison, untwisted PCF generates a Stokes signal with an unpredictable polarization state, owing to its high sensitivity to external perturbations. We use chiral PCF to realize a circularly polarized continuous-wave Brillouin laser. The results pave the way for a new generation of stable circularly polarized SBS systems with applications in quantum manipulation, optical tweezers, optical gyroscopes, and fiber sensors.