To operate at a building scale, service robots must perform very long-horizon mobile manipulation tasks by navigating to different rooms, accessing different floors, and interacting with a wide and unseen range of everyday objects. We refer to these tasks as Building-wide Mobile Manipulation. To tackle these inherently long-horizon tasks, we introduce BUMBLE, a unified Vision-Language Model (VLM)-based framework integrating open-world RGBD perception, a wide spectrum of gross-to-fine motor skills, and dual-layered memory. Our extensive evaluation (90+ hours) indicates that BUMBLE outperforms multiple baselines in long-horizon building-wide tasks that require sequencing up to 12 ground truth skills spanning 15 minutes per trial. BUMBLE achieves 47.1% success rate averaged over 70 trials in different buildings, tasks, and scene layouts from different starting rooms and floors. Our user study demonstrates 22% higher satisfaction with our method than state-of-the-art mobile manipulation methods. Finally, we demonstrate the potential of using increasingly-capable foundation models to push performance further. For more information, see: robin-lab.cs.utexas.edu/BUMBLE