Sim was born in Edinburgh, the youngest child and second son of Isabella (née McIntyre) and Alexander Sim. His mother moved to Edinburgh as a teenager from Eigg, one of the Small Isles in the Hebrides, and was a native Gaelic speaker. His father was a Justice of the Peace and a successful tailor with a business on Lothian Road. Sim was educated at Bruntsfield Primary school, James Gillespie's High School and George Heriot's School. He worked – probably part-time – in his father's shop and then for the men's outfitters Gieve's, displaying no talent for the retail trade. In 1918 he was admitted to the University of Edinburgh to study analytical chemistry, but was called up for army training.
After the end of the First World War in November 1918, Sim was released from military service. On his return home he told his family that he did not intend to resume his studies at the university, but instead would become an actor. His announcement was so badly received that he left the parental home, and spent about a year in the Scottish Highlands with a group of itinerant jobbing workers. Returning to Edinburgh, he took a post in the burgh assessor's office. In his spare time, he joined poetry reading classes, winning the gold medal for verse speaking at the Edinburgh Music Festival. This led to his engagement to teach elocution at a further education college in Dalry, Edinburgh. He held this post from 1922 to 1924. After taking an advanced training course in his subject, in 1925 he successfully applied to the University of Edinburgh for the post of Fulton Lecturer in Elocution, which he held for five years.