Autonomous fabric manipulation is a longstanding challenge in robotics, but evaluating progress is difficult due to the cost and diversity of robot hardware. Using Reach, a cloud robotics platform that enables low-latency remote execution of control policies on physical robots, we present the first systematic benchmarking of fabric manipulation algorithms on physical hardware. We develop 4 novel learning-based algorithms that model expert actions, keypoints, reward functions, and dynamic motions, and we compare these against 4 learning-free and inverse dynamics algorithms on the task of folding a crumpled T-shirt with a single robot arm. The entire lifecycle of data collection, model training, and policy evaluation is performed remotely without physical access to the robot workcell. Results suggest a new algorithm combining imitation learning with analytic methods achieves 84% of human-level performance on the folding task. See https://sites.google.com/berkeley.edu/cloudfolding for all data, code, models, and supplemental material.