In this paper, we propose a self-supervised single-view pixel-level accurate depth estimation network, called PLADE-Net. The PLADE-Net is the first work that shows unprecedented accuracy levels, exceeding 95\% in terms of the $\delta^1$ metric on the challenging KITTI dataset. Our PLADE-Net is based on a new network architecture with neural positional encoding and a novel loss function that borrows from the closed-form solution of the matting Laplacian to learn pixel-level accurate depth estimation from stereo images. Neural positional encoding allows our PLADE-Net to obtain more consistent depth estimates by letting the network reason about location-specific image properties such as lens and projection distortions. Our novel distilled matting Laplacian loss allows our network to predict sharp depths at object boundaries and more consistent depths in highly homogeneous regions. Our proposed method outperforms all previous self-supervised single-view depth estimation methods by a large margin on the challenging KITTI dataset, with unprecedented levels of accuracy. Furthermore, our PLADE-Net, naively extended for stereo inputs, outperforms the most recent self-supervised stereo methods, even without any advanced blocks like 1D correlations, 3D convolutions, or spatial pyramid pooling. We present extensive ablation studies and experiments that support our method's effectiveness on the KITTI, CityScapes, and Make3D datasets.