Conversational question answering (CQA) facilitates an incremental and interactive understanding of a given context, but building a CQA system is difficult for many domains due to the problem of data scarcity. In this paper, we introduce a novel method to synthesize data for CQA with various question types, including open-ended, closed-ended, and unanswerable questions. We design a different generation flow for each question type and effectively combine them in a single, shared framework. Moreover, we devise a hierarchical answerability classification (hierarchical AC) module that improves quality of the synthetic data while acquiring unanswerable questions. Manual inspections show that synthetic data generated with our framework have characteristics very similar to those of human-generated conversations. Across four domains, CQA systems trained on our synthetic data indeed show good performance close to the systems trained on human-annotated data.