Monocular depth estimators either require explicit scale supervision through auxiliary sensors or suffer from scale ambiguity, which renders them difficult to deploy in downstream applications. A possible source of scale is the sizes of objects found in the scene, but inaccurate localization makes them difficult to exploit. In this paper, we introduce a novel scale-aware monocular depth estimation method called StableCamH that does not require any auxiliary sensor or supervision. The key idea is to exploit prior knowledge of object heights in the scene but aggregate the height cues into a single invariant measure common to all frames in a road video sequence, namely the camera height. By formulating monocular depth estimation as camera height optimization, we achieve robust and accurate unsupervised end-to-end training. To realize StableCamH, we devise a novel learning-based size prior that can directly convert car appearance into its dimensions. Extensive experiments on KITTI and Cityscapes show the effectiveness of StableCamH, its state-of-the-art accuracy compared with related methods, and its generalizability. The training framework of StableCamH can be used for any monocular depth estimation method and will hopefully become a fundamental building block for further work.