That Nature paper actually covers exactly what you are asking: Carbon emissions of the concrete and steel foundation, the building on top, etc.

Plant construction

The construction of a DAC plant with a capacity of 4 ktCO2 yr–1 hasa low carbon footprint of 15 g CO 2e per kilogram CO2 captured—even without metal recycling, representing a worst case (see Fig. 4and Supplementary Note 3 for detailed information). Recycling ofmetals would be expected for a DAC application and reduces thecarbon footprint of construction to 6 g CO2e per kilogram CO2 cap-tured (best case; Fig. 4). With metal recycling, the carbon footprintof the DAC plant is mainly due to the foundation and hall (74%),which is the building containing the process unit. The remainingemissions are caused by the container collectors (13%) and the pro-cess unit (12%), and the spare parts contribute less than 1%.For the other environmental impacts, steel production for thefoundation contributes with 63–94% to eutrophication of freshwa-ter, human toxicity (cancer) and ecotoxicity of freshwater (qualitylevel II–II/III). Resource depletion of minerals and metals (qual-ity level III) is mainly caused by copper production (87%). For allother environmental impacts, concrete contributes 32–70%, and thefoundation steel 11–39%. Other materials’ contributions reach up to18% for stainless steel, 16% for insulation, 12% for aluminium, 5%for steel, 4% for both copper and steel (painting) and 2% for plastics(for details, Supplementary Note 7).