Accurate liver and lesion segmentation from computed tomography (CT) images are highly demanded in clinical practice for assisting the diagnosis and assessment of hepatic tumor disease. However, automatic liver and lesion segmentation from contrast-enhanced CT volumes is extremely challenging due to the diversity in contrast, resolution, and quality of images. Previous methods based on UNet for 2D slice-by-slice or 3D volume-by-volume segmentation either lack sufficient spatial contexts or suffer from high GPU computational cost, which limits the performance. To tackle these issues, we propose a novel context-aware PolyUNet for accurate liver and lesion segmentation. It jointly explores structural diversity and consecutive t-adjacent slices to enrich feature expressive power and spatial contextual information while avoiding the overload of GPU memory consumption. In addition, we utilize zoom out/in and two-stage refinement strategy to exclude the irrelevant contexts and focus on the specific region for the fine-grained segmentation. Our method achieved very competitive performance at the MICCAI 2017 Liver Tumor Segmentation (LiTS) Challenge among all tasks with a single model and ranked the $3^{rd}$, $12^{th}$, $2^{nd}$, and $5^{th}$ places in the liver segmentation, lesion segmentation, lesion detection, and tumor burden estimation, respectively.