Bayesian reward learning from demonstrations enables rigorous safety and uncertainty analysis when performing imitation learning. However, Bayesian reward learning methods are typically computationally intractable for complex control problems. We propose Bayesian Reward Extrapolation (Bayesian REX), a highly efficient Bayesian reward learning algorithm that scales to high-dimensional imitation learning problems by pre-training a low-dimensional feature encoding via self-supervised tasks and then leveraging preferences over demonstrations to perform fast Bayesian inference. Bayesian REX can learn to play Atari games from demonstrations, without access to the game score and can generate 100,000 samples from the posterior over reward functions in only 5 minutes on a personal laptop. Bayesian REX also results in imitation learning performance that is competitive with or better than state-of-the-art methods that only learn point estimates of the reward function. Finally, Bayesian REX enables efficient high-confidence policy evaluation without having access to samples of the reward function. These high-confidence performance bounds can be used to rank the performance and risk of a variety of evaluation policies and provide a way to detect reward hacking behaviors.