The goal of creating machines that autonomously perform useful work in a safe, robust and intelligent manner continues to motivate robotics research. Achieving this autonomy requires capabilities for understanding the environment, physically interacting with it, predicting the outcomes of actions and reasoning with this knowledge. Such intelligent physical interaction was at the centre of early robotic investigations and remains an open topic. In this paper, we build on the fruit of decades of research to explore further this question in the context of autonomous construction in unknown environments with scarce resources. Our scenario involves a miniature mobile robot that autonomously maps an environment and uses cubes to bridge ditches and build vertical structures according to high-level goals given by a human. Based on a "real but contrived" experimental design, our results encompass practical insights for future applications that also need to integrate complex behaviours under hardware constraints, and shed light on the broader question of the capabilities required for intelligent physical interaction with the real world.