He was born to a Jewish family in Bendery, Bessarabia, in the Russian Empire, and studied at the St Vladimir University in Kiev, leaving the USSR after the October Revolution and settling in the UK. He had positions at University College London and at the London School of Economics, before being appointed Professor of Economic History at the University of Cambridge, from 1937. He was known worldwide as an economic historian of medieval Europe. Hobsbawn notes he was one of the best lecturers at Cambridge, adding, "Though passionately anti-communist, Postan was the only man in Cambridge who knew Marx, Weber, Sombart and the rest of the great central and East Europeans, and took their work sufficiently seriously to expound and criticize it."
He married first the historian Eileen Power. After her death, he married Lady Cynthia Rosalie Keppel, daughter of the 9th Earl of Albemarle, with whom he had two sons.