Hugo's uncle, Ignacy Steinhaus [pl], was a lawyer and an activist in the Koło Polskie (Polish Circle), and a deputy to the Galician Diet, the regional assembly of the Kingdom of Galicia and Lodomeria.

[3] At the start of World War I Steinhaus returned to Poland and served in Józef Piłsudski's Polish Legion, after which he lived in Kraków.

According to Steinhaus, during the experience of this period, he "acquired an insurmountable physical disgust in regard to all sorts of Soviet administrators, politicians and commissars"[A] During the interwar period and the time of the Soviet occupation, Steinhaus contributed ten problems to the famous Scottish Book, including the last one, recorded shortly before Lwów was captured by the Nazis in 1941, during Operation Barbarossa.

[4] Steinhaus, because of his Jewish background, spent the Nazi occupation in hiding, first among friends in Lwów, then in the small towns of Osiczyna, near Zamość and Berdechów, near Kraków.

Worried about the possibility of imminent death if captured by Germans, Steinhaus, without access to any scholarly material, reconstructed from memory and recorded all the mathematics he knew, in addition to writing other voluminous memoirs, of which only a little part has been published.

[8] Also while in hiding, and cut off from reliable news on the course of the war, Steinhaus devised a statistical means of estimating for himself the German casualties at the front based on sporadic obituaries published in the local press.

Although initially he had doubts, he turned down offers for faculty positions in Łódź and Lublin and made his way to the city where he began teaching at University of Wrocław.

[4] Unlike his student, Stefan Banach, who tended to specialize narrowly in the field of functional analysis, Steinhaus made contributions to a wide range of mathematical sub-disciplines, including geometry, probability theory, functional analysis, theory of trigonometric and Fourier series as well as mathematical logic.

[3][4] He also wrote in the area of applied mathematics and enthusiastically collaborated with engineers, geologists, economists, physicians, biologists and, in Kac's words, "even lawyers".

[8] Probably his most notable contribution to functional analysis was the 1927 proof of the Banach–Steinhaus theorem, given along with Stefan Banach, which is now one of the fundamental tools in this branch of mathematics.

His interest in games led him to propose an early formal definition of a strategy, anticipating John von Neumann's more complete treatment of a few years later.

[4] While in hiding during World War II, Steinhaus worked on the fair cake-cutting problem: how to divide a heterogeneous resource among several people with different preferences such that every person believes he received a proportional share.

[14] Steinhaus had full command of several foreign languages and was known for his aphorisms, to the point that a booklet of his most famous ones in Polish, French and Latin has been published posthumously.