Open-domain tabular question answering (TQA) aims to answer natural language queries over a data lake that contains a large corpus of heterogeneous tables. Unlike closed-domain TQA, this task operates without pre-provided tables, making it particularly challenging. Existing methods usually first retrieve relevant tables and then answer the question based on the retrieved tables. However, it faces challenges in effectively and efficiently preparing tables from the low-quality, heterogeneous data lake and in operationalizing table information. To address these limitations, we present ArrTab (Agentic retrieval and reasoning on Tables) for open-domain TQA. ArrTab leverages MuRe, a self-supervised Multi-table Retriever trained on questions synthesized by translating table-derived SQL queries into natural language via large language models. Based on MuRe, ArrTab proposes a multi-agent system for open-domain TQA that consists of the Table Retriever Agent, the Table Integration Agent, the Information Extraction Agent and the Answer Agent. Furthermore, to enhance accuracy, a lazy execution strategy is adopted where operations are described in natural language but executed only during the information extraction stage, allowing for globally consistent reasoning. Extensive experiments on real-world datasets demonstrate the superiority of ArrTab.