The pursuit of human-like conversational agents has long been guided by the Turing test. For modern speech-to-speech (S2S) systems, a critical yet unanswered question is whether they can converse like humans. To tackle this, we conduct the first Turing test for S2S systems, collecting 2,968 human judgments on dialogues between 9 state-of-the-art S2S systems and 28 human participants. Our results deliver a clear finding: no existing evaluated S2S system passes the test, revealing a significant gap in human-likeness. To diagnose this failure, we develop a fine-grained taxonomy of 18 human-likeness dimensions and crowd-annotate our collected dialogues accordingly. Our analysis shows that the bottleneck is not semantic understanding but stems from paralinguistic features, emotional expressivity, and conversational persona. Furthermore, we find that off-the-shelf AI models perform unreliably as Turing test judges. In response, we propose an interpretable model that leverages the fine-grained human-likeness ratings and delivers accurate and transparent human-vs-machine discrimination, offering a powerful tool for automatic human-likeness evaluation. Our work establishes the first human-likeness evaluation for S2S systems and moves beyond binary outcomes to enable detailed diagnostic insights, paving the way for human-like improvements in conversational AI systems.