We present CloudGripper, an open source cloud robotics testbed, consisting of a scalable, space and cost-efficient design constructed as a rack of 32 small robot arm work cells. Each robot work cell is fully enclosed and features individual lighting, a low-cost Cartesian robot arm with an attached rotatable parallel jaw gripper and a dual camera setup for experimentation. The system design is focused on continuous operation and features a 10 Gbit/s network connectivity allowing for high throughput remote-controlled experimentation and data collection for robotic manipulation. Furthermore, CloudGripper is intended to form a community testbed to study the challenges of large scale machine learning and cloud and edge-computing in the context of robotic manipulation. In this work, we describe the mechanical design of the system, its initial software stack and evaluate the repeatability of motions executed by the proposed robot arm design. A local network API throughput and latency analysis is also provided. CloudGripper-Rope-100, a dataset of more than a hundred hours of randomized rope pushing interactions and approximately 4 million camera images is collected and serves as a proof of concept demonstrating data collection capabilities. A project website with more information is available at https://cloudgripper.org.