Dramatic cost reductions driven by private sector innovations have led to a rapid increase in the number of satellites in orbit and a corresponding surge in space-generated data. As this trend continues, transmitting large volumes of data to Earth for processing may become increasingly costly and challenging due to potential space-to-Earth link congestion and increased latency. Moreover, traditional ground station networks may face difficulties accommodating growing data flows and workloads because of capacity constraints, complex scheduling logistics, and restricted visibility windows, which can limit scalability. Space Data Centers (SDCs) -- software-driven, multi-tenant artificial intelligence-based service platforms capable of processing data in orbit to generate actionable insights for client satellites and ground users -- represent a promising approach to address these challenges. This article presents the architecture of a Low Earth Orbit SDC satellite constellation, considering orbital design, inter-satellite links and network topology, computational resource organization, and software service orchestration. We analyze the potential technical feasibility and economic viability of SDCs using forecasting models informed by technology roadmaps and illustrate the concept through Earth observation and lunar exploration use cases.