Tables are pervasive on the Web. Informative web tables range across a large variety of topics, which can naturally serve as a significant resource to satisfy user information needs. Driven by such observations, in this paper, we investigate an important yet largely under-addressed problem: Given millions of tables, how to precisely retrieve table cells to answer a user question. This work proposes a novel table cell search framework to attack this problem. We first formulate the concept of a relational chain which connects two cells in a table and represents the semantic relation between them. With the help of search engine snippets, our framework generates a set of relational chains pointing to potentially correct answer cells. We further employ deep neural networks to conduct more fine-grained inference on which relational chains best match the input question and finally extract the corresponding answer cells. Based on millions of tables crawled from the Web, we evaluate our framework in the open-domain question answering (QA) setting, using both the well-known WebQuestions dataset and user queries mined from Bing search engine logs. On WebQuestions, our framework is comparable to state-of-the-art QA systems based on knowledge bases (KBs), while on Bing queries, it outperforms other systems with a 56.7% relative gain. Moreover, when combined with results from our framework, KB-based QA performance can obtain a relative improvement of 28.1% to 66.7%, demonstrating that web tables supply rich knowledge that might not exist or is difficult to be identified in existing KBs.