Eshun was born in London, the younger brother of writer Kodwo Eshun. His family are Fante from Ghana, but although three years of his childhood were spent in Accra, for the most part he was brought up in London. He attended Kingsbury High School in North West London. and read politics and history at the London School of Economics. During his time there he edited both Features and Arts for the student newspaper The Beaver.
He sometimes appears as a critic on Saturday Review on BBC Radio 4. He appeared in 2009 in the television advertisements for Aviva (formerly Norwich Union). He also sometimes appears as a critic on The Review Show on BBC Two, as on 30 September 2011.