Varela was born in Santiago, Chile, the daughter of Leonor Palma-Keller, a massage therapist, and biologist and cognitive neuroscientist Francisco Varela García. She has a sister, Alejandra, and two brothers, Javier and Gabriel. Her mother is of French, Hungarian, and Syrian descent. When Varela was a child, her family fled Chile after the 1973 military coup d'état. In the early 1980s, her parents moved back to Chile while she remained in Paris where she studied acting at the "Niels Arestrup School", "L'École du Passage", and "Conservatoire Supérieur de Paris".