Grant was born Archibald Alec Leach on January 18, 1904, at 15 Hughenden Road in the northern Bristol, England suburb of Horfield. He was the second child of Elias James Leach and Elsie Maria Leach (née Kingdon). His father worked as a tailor's presser at a clothes factory, while his mother worked as a seamstress. His older brother John William Elias Leach died of tuberculous meningitis two days before his first birthday. Grant may have considered himself partly Jewish. He had an unhappy upbringing; his father was an alcoholic and his mother had clinical depression.
Grant's mother taught him song and dance when he was four, and she was keen on his having piano lessons. She occasionally took him to the cinema, where he enjoyed the performances of Charlie Chaplin, Chester Conklin, Fatty Arbuckle, Ford Sterling, Mack Swain, and Broncho Billy Anderson. He moved a short distance to 50 Berkeley Road at about age 4, and was sent to Bishop Road Primary School when he was 4 1/2. He lived at six addresses in Bristol, but said his time at Berkeley Road were his childhood "happiest days".
Grant's biographer Graham McCann claimed that his mother "did not know how to give affection and did not know how to receive it either". Biographer Geoffrey Wansell notes that his mother blamed herself bitterly for the death of Grant's brother John, and never recovered from it. Grant acknowledged that his negative experiences with his mother affected his relationships with women later in life. She frowned on alcohol and tobacco, and would reduce pocket money for minor mishaps. Grant attributed her behavior to overprotectiveness, fearing that she would lose him as she did John.
When Grant was nine, his father placed his mother in Glenside Hospital, a mental institution, and told him she had gone away on a "long holiday", later declaring that she had died. Grant grew up resenting his mother, particularly after being told she left the family. After she was institutionalised, Grant and his father moved into Grant's grandmother's home in Bristol. When Grant was ten, his father remarried and started a new family. Grant did not learn that his mother was still alive until he was 31, his father confessing to the lie shortly before his own death. Grant made arrangements for his mother to leave the institution in June 1935, shortly after he learned of her whereabouts. He visited her regularly, including after filming Gunga Din in October 1938.
Grant enjoyed the theater, particularly pantomimes at Christmas, which he attended with his father. He befriended a troupe of acrobatic dancers known as The Penders or the Bob Pender Stage Troupe. He subsequently trained as a stilt walker and began touring with them. Jesse Lasky was a Broadway producer at the time and saw Grant performing at the Wintergarten theater in Berlin around 1914.
In 1915, Grant won a scholarship to attend Fairfield Grammar School in Bristol, although his father could barely afford to pay for the uniform. He was quite capable in most academic subjects, but he excelled at sports, particularly fives, and his good looks and acrobatic talents made him a popular figure. He developed a reputation for mischief, and frequently refused to do his homework. A former classmate referred to him as a "scruffy little boy", while an old teacher remembered "the naughty little boy who was always making a noise in the back row and would never do his homework".
He spent his evenings working backstage in Bristol theatres, and at the age of 13, was responsible for the lighting for magician David Devant at the Bristol Empire in 1917. He began hanging around backstage at the theatre at every opportunity, and volunteered for work in the summer as a messenger boy and guide at the military docks in Southampton, to escape the unhappiness of his home life. The time spent at Southampton strengthened his desire to travel; he was eager to leave Bristol and tried to sign on as a ship's cabin boy, but he was too young.
On March 13, 1918, the 14-year-old Grant was expelled from Fairfield. Several explanations were given, including being discovered in the girls' lavatory and assisting two other classmates with theft in the nearby town of Almondsbury. Wansell claims that Grant had set out intentionally to get himself expelled from school to pursue a career in entertainment with the troupe, and he did rejoin Pender's troupe three days after being expelled. His father had a better-paying job in Southampton, and Grant's expulsion brought local authorities to Pender's door with questions about why he was living in Bristol and not with his father in Southampton. His father then co-signed a three-year contract between Grant and Pender that stipulated Grant's weekly salary, along with room and board, dancing lessons, and other training for his profession until age 18. There was also a provision in the contract for salary raises based on job performance.