Eshelby was born at Puddington, Cheshire, the son of Captain Alan Douglas Eshelby and Phoebe Mason Hutchinson. He was educated at St Cyprian's School, Eastbourne and was due to go to Charterhouse School but developed rheumatic fever and received his secondary education privately at home. At about this time the family moved to Manor House at Farrington Gurney, Somerset where his tutors were the village schoolmaster and a local clergyman. He relied extensively on self-instruction and obtained a place in the Physics Department of Bristol University and was awarded a first class honours in physics in 1937. He then worked in a research laboratory under H W B Skinner and W Sucksmith on magnetism and the soft X-ray spectra of solids.