Cloud computing has begun a transformation from using virtual machines to containers. Containers are attractive because multiple of them can share a single kernel, and add minimal performance overhead. Cloud providers leverage the lean nature of containers to run hundreds of them on a few cores. Furthermore, containers enable the serverless paradigm, which leads to the creation of short-lived processes. In this work, we identify that containerized environments create page translations that are extensively replicated across containers in the TLB and in page tables. The result is high TLB pressure and redundant kernel work during page table management. To remedy this situation, this paper proposes BabelFish, a novel architecture to share page translations across containers in the TLB and in page tables. We evaluate BabelFish with simulations of an 8-core processor running a set of Docker containers in an environment with conservative container co-location. On average, under BabelFish, 53% of the translations in containerized workloads and 93% of the translations in serverless workloads are shared. As a result, BabelFish reduces the mean and tail latency of containerized data-serving workloads by 11% and 18%, respectively. It also lowers the execution time of containerized compute workloads by 11%. Finally, it reduces serverless function bring-up time by 8% and execution time by 10%-55%.