Zach, the youngest of sixteen children in a working-class family, was born in the medieval town of Slaný, northwest of Prague. At age fifteen, he left home to earn his living in Prague by painting theater posters and singing in pubs. He apprenticed with several well-known painters, but it was the work of the sculptor Zdeněk Pešánek (with whom Zach never studied directly) that deeply affected him. Pešánek created illuminated kinetic sculptures designed as outdoor advertising apparati for electric substations and commercial buildings in Prague. The intersection of art, technology, and civic space became an abiding interest for Zach.
While Zach was in New York City in 1938, decorating the Czech Pavilion at the 1939 World's Fair, the Nazis invaded Czechoslovakia. Zach never returned. He spent the 1940s in Brazil, joining the international art scene in Rio de Janeiro. He had large exhibitions of his paintings and drawings at the National Museum of Fine Arts in 1943 and the Institute of Brazilian Architects in 1948. He also worked as a commercial artist in Rio.