Reinforcement learning (RL) can in principle let robots automatically adapt to new tasks, but current RL methods require a 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. Fast adaptation is crucial because prohibitively large number of on-robot trials will potentially damage hardware pieces. Additionally, effective adaptation is also feasible in that experience among different insertion applications can be largely leveraged by each other. In this setting, we address two specific challenges when applying meta-learning. First, conventional meta-RL algorithms require lengthy online meta-training. 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, 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/offline-metarl-insertion.