Kobozev was born in a wealthy family of a Kharkov lawyer. His father, Ivan Josefovich Kobozev (1874, Kharkov – 1943, Moscow) graduated from Kharkov Imperial University. His mother, Sophia Adolfovna Feist (d. 1952) was a granddaughter of the German-born Taganrog watchmaker Franz Feist (1805–1888). Her family was Lutheran. Her father, Adolf Feist, was first a teacher of German; in 1891 he became a member of the board of Kharkov Land Bank. His mother's aunt, Maria Feist, was a sweetheart of Alexander Chekhov.
Nikolay's grandfather, Josef Alekseevich Kobozev (1846, Belgorod – July 18, 1901, Kharkov) moved to Kharkov in the 1860s where he got engaged in textile and flour trade. In 1889 he became a board member of the Second Kharkov Society of Mutual Credit. In 1892 he was elected to the Kharkov city duma.
According to independent researcher Igor Maslenkov, Kobozevs earliest ancestor was a serf peasant of a Belgorod boyar scion Artyom Pischyulin, settled at the village of Melehovo. His grandson, Dmitry Kosmin syn Kobyzev (1697–1752) became a merchant in Belgorod. His wife, Agrafena Fedotovna Maslova (1689–1770) was the daughter of a local poor nobleman. Their son, Stepan, was a merchant too; he was married to the daughter of another Belgorod merchant, Stephanida Rodionovna Dubinina. Stepans brother, Ivan Dmitrievich Kobozev, was the salt head in Belgorod since 1778. Yakov Stepanovich Kobozev was a Belgorod merchant and ratman. The latter's son, Nikolay Yakovlevich (1781–1834), was a merchant too, but his son, Aleksey Nikolayevich (1804–?) left the merchant guild for the status of an ordinary burger in 1858.
In 1903 his father moved to Moscow, where he worked as a lawyer () at the Administration of Moscow Vindavo-Rybinsk Railroad. After the October Revolution of 1917, he served as a lawyer at the Ministry of Transport.
Nikolays brother, Vsevolod (1905–1939), was an engineer at the Soviet Ministry of Transport, the head of the electrification department. He was married to the daughter of Vissarion Karandeev, the professor of Moscow higher womens courses. For successful electrification of the Yaroslavl railways Stalin granted him with a golden watch. But in 1937 he was arrested and, in 1939, executed by shooting.
His other brother, Boris was a musician; he died in 1918 from Spanish flu. He also had a sister named Inna, who was seriously ill for most of her life.
Kobozev was married to Esther (Ekaterina in Russian Orthodoxy) Efimovna Halbreich of Jewish background. In 1946 they had a son named Aleksey (1946–2015), who had a daughter. His brother's lineage is continued in the male line.
Since early childhood, Kobozev was ill with poliomielitis. Since the 1920s, he had had a number of serious diseases, including complications of the polio. At some point in his life, he was wheelchaired and had to meet with his students at home. He had long periods of staying at hospitals or was down in bed recovering. In 1973 he was completely bed-bound. His wife had stayed loyal and supportive throughout his life.