One variation of a method includes: ingesting an air sample captured during an air capture period at a target location for collection of a first mixture including carbon dioxide and a first concentration of impurities; conveying the first mixture through a liquefaction unit to generate a second mixture including carbon dioxide and a second concentration of impurities less than the first concentration of impurities; in a methanation reactor, mixing the second mixture with hydrogen to generate a first hydrocarbon mixture comprising a third concentration of impurities comprising nitrogen, carbon dioxide, and hydrogen; conveying the first hydrocarbon mixture through a separation unit configured to remove impurities from the first hydrocarbon mixture to generate a second hydrocarbon a fourth concentration of impurities less than the third concentration of impurities; and depositing the second hydrocarbon mixture in a diamond reactor containing a set of diamond seeds to generate a first set of diamonds.