Bárány became a doctor of philosophy in theoretical physics at Uppsala University in 1973 and later became university doctor at the same university. During the 1980s he worked at the Research Institute for Atomic Physics in Stockholm. Since parts of the institute was transferred to Stockholm University and renamed as the Manne Sieghbahn Laboratory, Bárány became a professor there.
Bárány has worked at the Kungliga Vetenskapsakademien.
He was an editor at Physica Scripta between 1988 and 1996 and secretary for the Nobel committee in physics between 1989 and 2003. Bárány has been the chairman of the Kungliga Vetenskaps Societeten in Uppsala since 1994, and of the Kungliga Vetenskapsakademien since 2005.
He was earlier the chairman of the Nobel Museum. Bárány was given the honour of being the host of the radio show Sommar i P1 at Sveriges Radio P1 on 30 June 2014.
Royal Swedish Academy of Sciences.