Despite recent progress on computer vision and natural language processing, developing a machine that can understand video story is still hard to achieve due to the intrinsic difficulty of video story. Moreover, researches on how to evaluate the degree of video understanding based on human cognitive process have not progressed as yet. In this paper, we propose a novel video question answering (Video QA) task, DramaQA, for a comprehensive understanding of the video story. The DramaQA focuses on two perspectives: 1) Hierarchical QAs as an evaluation metric based on the cognitive developmental stages of human intelligence. 2) Character-centered video annotations to model local coherence of the story. Our dataset is built upon the TV drama "Another Miss Oh" and it contains 17,983 QA pairs from 23,928 various length video clips, with each QA pair belonging to one of four difficulty levels. We provide 217,308 annotated images with rich character-centered annotations, including visual bounding boxes, behaviors and emotions of main characters, and coreference resolved scripts. Additionally, we suggest Multi-level Context Matching model which hierarchically understands character-centered representations of video to answer questions. We release our dataset and model publicly for research purposes, and we expect our work to provide a new perspective on video story understanding research.