Reinforcement learning (RL) is capable of sophisticated motion planning and control for robots in uncertain environments. However, state-of-the-art deep RL approaches typically lack safety guarantees, especially when the robot and environment models are unknown. To justify widespread deployment, robots must respect safety constraints without sacrificing performance. Thus, we propose a Black-box Reachability-based Safety Layer (BRSL) with three main components: (1) data-driven reachability analysis for a black-box robot model, (2) a trajectory rollout planner that predicts future actions and observations using an ensemble of neural networks trained online, and (3) a differentiable polytope collision check between the reachable set and obstacles that enables correcting unsafe actions. In simulation, BRSL outperforms other state-of-the-art safe RL methods on a Turtlebot 3, a quadrotor, a trajectory-tracking point mass, and a hexarotor in wind with an unsafe set adjacent to the area of highest reward.