Naturally-occurring information-seeking questions often contain questionable assumptions -- assumptions that are false or unverifiable. Questions containing questionable assumptions are challenging because they require a distinct answer strategy that deviates from typical answers to information-seeking questions. For instance, the question "When did Marie Curie discover Uranium?" cannot be answered as a typical when question without addressing the false assumption "Marie Curie discovered Uranium". In this work, we propose (QA)$^2$ (Question Answering with Questionable Assumptions), an open-domain evaluation dataset consisting of naturally-occurring search engine queries that may or may not contain questionable assumptions. To be successful on (QA)$^2$, systems must be able to detect questionable assumptions and also be able to produce adequate responses for both typical information-seeking questions and ones with questionable assumptions. We find that current models do struggle with handling questionable assumptions -- the best performing model achieves 59% human rater acceptability on abstractive QA with (QA)$^2$ questions, leaving substantial headroom for progress.