Abram Molodetskii was born in 1923 in Odessa, the second of four children. His father Zelman was a tailor. His business was going badly, and so, in 1929, he joined a Jewish agricultural colony in the district of Kalinindorf in southern Ukraine. However, he was not successful there, either, and in 1932 the family returned to Odessa. Abram attended a Yiddish school, but after the 7th grade he was transferred to a Ukrainian one. In 1937, he dropped out of school and began to attend a technikum [college] for electrical communication; a year later, he transferred to the Odessa technikum of credit and economics. On June 22, 1941, the day of the outbreak of the Soviet-German War, Molodetskii had his final examination at the technikum. On July 23, 1941, youths of premilitary age were to be evacuated from Odessa eastward. However, the boat on which Abram was sailing was attacked by Romanian military aircraft, and some of the young people aboard were killed. Abram survived; he arrived in Dnepropetrovsk (present-day Dnipro, Ukraine), where he received a rifle and a military uniform, and was sent to dig trenches. A little later, he and his comrades were ordered to repel the German forces advancing in the area. There, Molodetskii was taken prisoner. In the first camp, having no documents on his person, he was registered as Andrei Molodetskii, a Ukrainian from the village of Kryve Ozero (north of Odessa).
The Germans herded the POWs taken in the area of Dnepropetrovsk westward, moving them from one camp to another. In late September, when the POWs reached the Odessa region, Molodetskii, as an alleged Ukrainian from a local village, was released. While meandering across the area, he stopped for a while in a Jewish agricultural colony that was still operating, and managed to escape from it on the night of the Nazi massacre of the colonists. In December 1941, he came to Odessa, hoping to find his relatives – but, having failed to do so, he went back to the countryside. In early 1942, he witnessed the mass murder of a group of Jews who had been driven from Odessa. In 1943, Molodetskii was hired as a farmhand on a farm of German colonists (whose ancestors had settled in the Odessa region in the early 19th century). In the fall of 1943, when the German colonists were evacuated from the area, Abram-Andrei fled eastward from the colony. In the spring of 1944, he encountered the Red Army.
Unexpectedly, instead of being drafted, Abram Molodetskii, as the graduate of an economic technikum, was sent to reestablish a bank in the town of Kominternivske (present-day Dobroslav) near Odessa. This done, he worked as the manager of the bank for about two months. Then, in May 1944, he volunteered to enlist in the Red Army. Molodetskii (who initially kept his alias Andrei even in the Red Army) was assigned to a communication platoon of the 51st Rifle Regiment, in the rank of second sergeant. He saw action in Bessarabia (Moldova), Romania, Hungary, Yugoslavia, and Austria. Later in 1944, he was transferred to an air regiment. At the same time, he restored his Jewish identity and officially changed his first name back to "Abram". Abram Molodetskii was awarded the medal "For Courage" (in Bessarabia, August 1944) and the Order of Glory, 3rd class, for the operation at Lake Balaton. Molodetskii deliberately refused to train for an officer rank, since he wished to conceal his stay in German captivity. Therefore, he ended the war in the rank of senior sergeant.
Abram Molodetskii was discharged from the army in 1948. He was reunited with his family in Tashkent, Uzbekistan. During his stay in Uzbekistan, Molodetskii was denounced and arrested by the MGB (the predecessor of the KGB) for failing to disclose the fact that he had been in German captivity. He was to be sentenced to 10 years of hard labor, but his father Zelman bribed his investigator, and Abram was released. That same year, Molodetskii became married. The wedding was a traditional ceremony, and was held at a synagogue. He had two sons.
In 1993, after the death of his wife, Abram Molodetskii immigrated to Israel, where he lived in Kiryat Ata. He died in 2011.