Born in Dunedin, New Zealand, he studied in that country but obtained his PhD in the United Kingdom at the University of Cambridge.

Later, he returned in New Zealand where he helped with the development of the Meteorological Office, becoming its director in 1939, just before the start of World War II.

In 1924, he graduated MSc in mathematics and physics, writing a thesis about the equipment used by Professor Robert Jack in his experimental radio broadcasts of 1921 and 1922.

Sir Ernest Rutherford assigned him the study of the propagation of radio waves through what became known later as the ionosphere under the supervision of Edward Victor Appleton.

In 1935, while in the United Kingdom, Barnett was appointed to the New Zealand Meteorological Office (then a branch of the DSIR, now MetService) to develop aviation services.