Radii Birbraier was born in 1923 in Moscow. His father, Moisei Birbraier, was an engineer. In 1930 Radii's father was sent to England to purchase electrical equipment for the USSR, within the framework of the joint Soviet-British Arcos Company. Since Moisei took his family with him and they remained for about three years, Radii had the opportunity to study in an English school and English became his second language. In 1933, the Birbraiers returned to the Soviet Union, settling in Leningrad. In 1937, Moisei Birbraier was arrested and disappeared into the Gulag. Radii and his mother were expelled from Leningrad to northeastern Kazakhstan, where his mother worked as a book-keeper for a village hospital.
With the beginning of the Soviet-German war in 1941, Radii was called up to the Red Army. However, instead of being assigned to army service, he was sent to the coal mines of Kuzbass (Siberia), where he worked as a driller and a blaster and, later, as a mucker. In January 1942, Birbraier was assigned to active military service in a separate engineering battalion and, later, was deployed in Ukraine. From the end of 1942 to the beginning of 1944, his battalion fought across the whole of Ukraine from Kharkov in the northeast to the Dniester River in the southwest. Subsequently, he ascribed his survival during the year and a half of fighting in Ukraine to the fact that his engineering service was less dangerous than other kinds of service and that his unit was rarely on the front lines. However, in October 1943 his battalion was involved in the forced crossing of the Dnieper. Birbraier was wounded internally and, therefore, hospitalized. For his role in the fighting, he was promoted to the rank of corporal.
In August 1944, Birbraier was sent to the Moscow Military Engineering School. After graduation, he returned to his battalion as a second lieutenant. His battalion was deployed near Budapest. He received one medal during the war. After the end of the war he was awarded the Order of the Red Star.
Birbraier continued his military service until 1952. After his retirement, he became a mathematician. From 1959 on, he worked at the Institute for Construction and Road Construction in Ust-Kamennogorsk (later The Technical University of East Kazakhstan). Birbraier was one of the pioneers of computer sciences in the Soviet Union.
In 1997 Birbraier immigrated to Israel, where he became active in organizations of World War II veterans.