Data processing systems are increasingly deployed in the cloud. While monolithic systems run fully on virtual servers, recent systems embrace cloud infrastructure and utilize the disaggregation of compute and storage to scale them independently. The introduction of serverless compute services, such as AWS Lambda, enables finer-grained and elastic scalability within these systems. Prior work shows the viability of serverless infrastructure for scalable data processing yet also sees limitations due to variable performance and cost overhead, in particular for networking and storage. In this paper, we perform a detailed analysis of the performance and cost characteristics of serverless infrastructure in the data processing context. We base our analysis on a large series of micro-benchmarks across different compute and storage services, as well as end-to-end workloads. To enable our analysis, we propose the Skyrise serverless evaluation platform. For the widely used serverless infrastructure of AWS, our analysis reveals distinct boundaries for performance variability in serverless networks and storage. We further present cost break-even points for serverless compute and storage. These insights provide guidance on when and how serverless infrastructure can be efficiently used for data processing.