Bloch was born in Zürich, Switzerland to Jewish parents Gustav and Agnes Bloch. Gustav Bloch, his father, was financially unable to attend University and worked as a wholesale grain dealer in Zürich. Gustav moved to Zürich from Moravia in 1890 to become a Swiss citizen. Their first child was a girl born in 1902 while Felix was born three years later.
Bloch entered public elementary school at the age of six and is said to have been teased, in part because he "spoke Swiss German with a somewhat different accent than most members of the class". He received support from his older sister during much of this time, but she died at the age of twelve, devastating Felix, who is said to have lived a "depressed and isolated life" in the following years. Bloch learned to play the piano by the age of eight and was drawn to arithmetic for its "clarity and beauty". Bloch graduated from elementary school at twelve and enrolled in the Cantonal Gymnasium in Zürich for secondary school in 1918. He was placed on a six-year curriculum here to prepare him for University. He continued his curriculum through 1924, even through his study of engineering and physics in other schools, though it was limited to mathematics and languages after the first three years. After these first three years at the Gymnasium, at age fifteen Bloch began to study at the Eidgenössische Technische Hochschule (ETHZ), also in Zürich. Although he initially studied engineering he soon changed to physics. During this time he attended lectures and seminars given by Peter Debye and Hermann Weyl at ETH Zürich and Erwin Schrödinger at the neighboring University of Zürich. A fellow student in these seminars was John von Neumann. 
Bloch graduated in 1927, and was encouraged by Debye to go to Leipzig to study with Werner Heisenberg. Bloch became Heisenberg's first graduate student, and gained his doctorate in 1928. His doctoral thesis established the quantum theory of solids, using waves to describe electrons in periodic lattices.
On March 14, 1940, Bloch married Lore Clara Misch (1911–1996), a fellow physicist working on X-ray crystallography, whom he had met at an American Physical Society meeting. They had four children, twins George Jacob Bloch and Daniel Arthur Bloch (born January 15, 1941), son Frank Samuel Bloch (born January 16, 1945), and daughter Ruth Hedy Bloch (born September 15, 1949).