He was co-editor of the collected works of Hassler Whitney. Eells's doctoral students include Peter Štefan (1941 - 1978), Giorgio Valli (1960 - 1999) and Pierre de la Harpe (de). Eells was married since 1950 and had a son and three daughters.

Eells studied mathematics at Bowdoin College in Maine and earned his undergraduate degree in 1947. After graduation he spent one year teaching mathematics at Robert College in Istanbul and starting in 1948 was for two years an instructor at Amherst College in Amherst. Next he undertook graduate study at Harvard, where in 1954 he received his PhD under Hassler Whitney with thesis Geometric Aspects of Integration Theory. In the academic year 1955–1956 he was at the Institute for Advanced Study (and subsequently in 1962–1963, 1972–1973 and 1977). He taught at Columbia University for several years. In 1964 he became a full professor at Cornell University. In 1963 and in 1966–1967 he was in Cambridge and after a visit to the new mathematics department developed by Erik Christopher Zeeman at the University of Warwick Eells became a professor of mathematical analysis there in 1969. Eells organized many of the University of Warwick Symposia in mathematics. In 1986 he became the first Director of the mathematics section of the Abdus Salam International Centre for Theoretical Physics in Trieste; for six years he served as Director in addition to his appointment at the University of Warwick. In 1992 he retired and lived in Cambridge.

In 1970 he was an invited speaker at the International Mathematical Congress in Nice (On Fredholm manifolds with K. D. Elworthy).