Arcel was born and raised in Copenhagen. His mother, Libby Tata Arcel, is a psychologist from the Greek city Mytilene, on the island of Lesbos, while his father, Arne Arcel, is an architect from Denmark. His parents divorced when he was seven. His elder sister is actress Nastja Arcel. He attended Bernadotte School in Hellerup and secondary school at Øster Borgerdyd Gymnasium. He then enrolled at the National Film School of Denmark from where he graduated as film director in 2001. His graduation project, the short film Woyzeck's Last Symphony, won the Grand Prix at Clermont-Ferrand International Short Film Festival.
