Safe real-time control of robotic manipulators in unstructured environments requires handling numerous safety constraints without compromising task performance. Traditional approaches, such as artificial potential fields (APFs), suffer from local minima, oscillations, and limited scalability, while model predictive control (MPC) can be computationally expensive. Control barrier functions (CBFs) offer a promising alternative due to their high level of robustness and low computational cost, but these safety filters must be carefully designed to avoid significant reductions in the overall performance of the manipulator. In this work, we introduce an Operational Space Control Barrier Function (OSCBF) framework that integrates safety constraints while preserving task-consistent behavior. Our approach scales to hundreds of simultaneous constraints while retaining real-time control rates, ensuring collision avoidance, singularity prevention, and workspace containment even in highly cluttered settings or during dynamic motions. By explicitly accounting for the task hierarchy in the CBF objective, we prevent degraded performance across both joint-space and operational-space tasks, when at the limit of safety. We validate performance in both simulation and hardware, and release our open-source high-performance code and media on our project webpage, https://stanfordasl.github.io/oscbf/