The principles of natural protein engineering are obscured by overlapping functions and complexity accumulated through natural selection and evolution. Completely artificial proteins offer a clean slate on which to define and test these protein engineering principles, while recreating and extending natural functions. Here we introduce this method with the design of an oxygen transport protein, akin to human neuroglobin. Beginning with a simple and unnatural helix-forming sequence with just three different amino acids, we assembled a four-helix bundle, positioned histidines to bis-histidine ligate haems, and exploited helical rotation and glutamate burial on haem binding to introduce distal histidine strain and facilitate O2 binding. For stable oxygen binding without haem oxidation, water is excluded by simple packing of the protein interior and loops that reduce helical-interface mobility. O2 affinities and exchange timescales match natural globins with distal histidines, with the remarkable exception that O2 binds tighter than CO.