He received his early education in Steinheim and Hanau, and then entered the seminary in Mainz, where he studied philosophy and theology for four years. From 1935 to 1938 he studied at the University of Fribourg in Switzerland, from where he obtained his doctorate in philosophy, with a thesis entitled: Die Kreaturauffassung von Karl Barth, and his doctorate in theology, with a thesis on: Emil Brunners Lehre von der ursprünglichen Gottesebenbildlichkeit des Menschen.
He served as Bishop of Mainz from 1962 to 1982, and was elevated to the cardinalate in 1973. Volk was ordained to the priesthood by Bishop Ludwig Maria Hugo on 27 April 1927. He then did pastoral work in Mainz, serving as a chaplain in Alzey (1927–1931) and in Saint Ignaz (1932–1935).
Volk was an assistant of a vicariate in Gau-Odernheim from 1940 to 1941, when he was named Vicar of Nidda, where he would then serve as pastor from 1942 to 1945.
In 1943 Volk obtained his habilitation in dogmatic theology from the University of Münster, with a work entitled: Emil Brunners Lehre von dem Sinder. He then served as Assistant Professor (1945–1946) and Professor (1946–1962) of Theology, and rector (1954–1955) at the University of Münster.
He was raised to the rank of a Domestic Prelate of His Holiness on 25 February 1962. On 3 March 1962 Volk was elected Bishop of Mainz by the cathedral chapter, and confirmed as such by Pope John XXIII on 25 March 1962.
He received his episcopal consecration on the following 5 June from Archbishop Hermann Schäufele, with Bishops Carl Joseph Leiprecht and Joseph Reuß serving as co-consecrators, in the Cathedral of Mainz.
Bishop Volk then attended the Second Vatican Council until 1965. Pope Paul VI created him Cardinal-Priest of Ss. Fabiano e Venanzio a Villa Fiorelli in the consistory of 5 March 1973.
Volk was one of the cardinal electors who participated in the conclaves of August and October 1978, which selected Popes John Paul I and John Paul II respectively.
After twenty years of service, he resigned as Bishop of Mainz on his seventy-ninth birthday, 27 December 1982. Volk later lost the right to participate in any future papal conclaves upon reaching age 80 on 27 December 1983.
The Cardinal died after a long illness in Mainz, at age 84. He is buried in the Cathedral of Mainz.