Akhiezer was born in Cherykaw, Russian Empire in what is now Mogilev Region, Belarus. He studied radio engineering at Kyiv Polytechnic Institute in 1929–34. From 1934, he worked at the Ukrainian Institute of Physics and Technology in Kharkiv. With Isaak Pomeranchuk and under the supervision of Lev Landau, he studied light-light scattering and was awarded a Ph.D. in 1936.