Although deep convolutional networks have been widely studied for head and neck (HN) organs at risk (OAR) segmentation, their use for routine clinical treatment planning is limited by a lack of robustness to imaging artifacts, low soft tissue contrast on CT, and the presence of abnormal anatomy. In order to address these challenges, we developed a computationally efficient nested block self-attention (NBSA) method that can be combined with any convolutional network. Our method achieves computational efficiency by performing non-local calculations within memory blocks of fixed spatial extent. Contextual dependencies are captured by passing information in a raster scan order between blocks, as well as through a second attention layer that causes bi-directional attention flow. We implemented our approach on three different networks to demonstrate feasibility. Following training using 200 cases, we performed comprehensive evaluations using conventional and clinical metrics on a separate set of 172 test scans sourced from external and internal institution datasets without any exclusion criteria. NBSA required a similar number of computations (15.7 gflops) as the most efficient criss-cross attention (CCA) method and generated significantly more accurate segmentations for brain stem (Dice of 0.89 vs. 0.86) and parotid glands (0.86 vs. 0.84) than CCA. NBSAs segmentations were less variable than multiple 3D methods, including for small organs with low soft-tissue contrast such as the submandibular glands (surface Dice of 0.90).