Research infrastructures have been identified as an important source of greenhouse gas emissions of astronomical research. Based on a comprehensive inventory of 1,211 ground-based observatories and space missions, we assessed the evolution of the number of astronomical facilities and their carbon footprint from 1945 to 2022. We found that space missions dominate greenhouse gas emissions in astronomy, showing an important peak at the end of the 1960s, followed by a decrease that has turned again into a rise over the last decade. Extrapolating past trends, we predict that greenhouse gas emissions from astronomical facilities will experience no strong decline in the future, and may even rise substantially, unless research practices are changed. We demonstrate that a continuing growth in the number of operating astronomical facilities is not environmentally sustainable. These findings should motivate the astronomical community to reflect about the necessary evolutions that would put astronomical research on a sustainable path.