Thermal machines exploit interactions with multiple heat baths to perform useful tasks, such as work production and refrigeration. In the quantum regime, tasks with no classical counterpart become possible. Here, we consider the minimal setting for quantum thermal machines, namely two-qubit autonomous thermal machines that use only incoherent interactions with their environment, and investigate the fundamental resources needed to generate entanglement. Our investigation is systematic, covering different types of interactions, bosonic and fermionic environments, and different resources that can be supplied to the machine. We adopt an operational perspective in which we assess the nonclassicality of the generated entanglement through its ability to perform useful tasks such as Einstein-Podolsky-Rosen steering, quantum teleportation and Bell nonlocality. We provide both constructive examples of nonclassical effects and general no-go results that demarcate the fundamental limits in autonomous entanglement generation. Our results open up a path toward understanding nonclassical phenomena in thermal processes.