In many different settings, requests for service can arrive in near or true simultaneity with one another. This creates batches of arrivals to the underlying queueing system. In this paper, we study the staffing problem for the batch arrival queue. We show that batches place a dangerous and deceptive stress on services, requiring a high amount of resources and exhibiting a fundamentally larger tail in those demands. This uncovers a service regime in which a system with large batch arrivals may have low utilization but will still have non-trivial waiting. Methodologically, these staffing results follow from novel large batch and large batch-and-rate limits of the multi-server queueing model. In the large batch limit, we establish the first formal connection between general multi-server queues and storage processes, another family of stochastic models. By consequence, we show that the batch scaled queue length process is not asymptotically normal, and that, in fact, the fluid and diffusion-type limits coincide. Hence, the (safety) staffing of this system must be directly proportional to the batch size just to achieve a non-degenerate probability of wait. In exhibition of the existence and insights of this large batch regime, we apply our results to data on Covid-19 contact tracing in New York City. In doing so, we identify significant benefits produced by the tracing agency's decision to staff above national recommendations, and we also demonstrate that there may have been an opportunity to further improve the operation by optimizing the arrival pattern in the public health data pipeline.