Thermal and acoustic noises are crucial to the long-term stability of fiber lasers, as it introduces the fluctuation of optical path length on laser cavity, and hence imposing undesirable intensity noise and frequency drift, particularly for a random fiber laser with distributed Rayleigh scattering feedback from a long length fiber. In this Letter, we propose and demonstrate a thermal and acoustic noise insensitive Brillouin random fiber laser by utilizing the random feedback from a polarization-maintaining (PM) fiber-based random grating. A theoretical model is developed for the first time, to the best of our knowledge, to analyze environmental perturbation on the randomly induced refractive index modulation via a PM random grating. Both the theory and experiments show that the scattered optical intensity of the PM random fiber grating exhibits a weak dependence on the temperature fluctuation and the acoustic noise perturbation compared to that of the Rayleigh scattering from hundreds of meters of PM fibers, leading to the Brillouin random lasing radiation with a 20 dB relative intensity noise suppression in the frequency range from 10 Hz to 1 kHz.