Reinforcement learning (RL) can in principle make it possible for robots to automatically adapt to new tasks, but in practice current RL methods require a very large number of trials to accomplish this. In this paper, we tackle rapid adaptation to new tasks through the framework of meta-learning, which utilizes past tasks to learn to adapt, with a specific focus on industrial insertion tasks. We address two specific challenges by applying meta-learning in this setting. First, conventional meta-RL algorithms require lengthy online meta-training phases. We show that this can be replaced with appropriately chosen offline data, resulting in an offline meta-RL method that only requires demonstrations and trials from each of the prior tasks, without the need to run costly meta-RL procedures online. Second, meta-RL methods can fail to generalize to new tasks that are too different from those seen at meta-training time, which poses a particular challenge in industrial applications, where high success rates are critical. We address this by combining contextual meta-learning with direct online finetuning: if the new task is similar to those seen in the prior data, then the contextual meta-learner adapts immediately, and if it is too different, it gradually adapts through finetuning. We show that our approach is able to quickly adapt to a variety of different insertion tasks, learning how to perform them with a success rate of 100% using only a fraction of the samples needed for learning the tasks from scratch. Experiment videos and details are available at https://sites.google.com/view/oda-anon.