Buildings are responsible for about a quarter of global energy-related CO2 emissions. Consequently, the decarbonisation of the housing stock is essential in achieving net-zero carbon emissions. Global decarbonisation targets can be achieved through increased efficiency in using energy generated by intermittent resources. The paper presents a co-design framework for simultaneous optimal design and operation of residential buildings using Model Predictive Control (MPC). The framework is capable of explicitly taking into account operational constraints and pushing the system to its efficiency and performance limits in an integrated fashion. The optimality criterion minimises system cost considering time-varying electricity prices and battery degradation. A case study illustrates the potential of co-design in enhancing flexibility and self-sufficiency of a system operating under different conditions. Specifically, numerical results from a low-fidelity model show substantial carbon emission reduction and bill savings compared to an a-priori sizing approach.