Eli Carr was born around 1836 in Albemarle County, Virginia. He was a slave of William Piper, a man who lived prior to the war in Saline County, Missouri, and owned three slaves. Carr worked as a farmer prior to the war, often breaking hemp and teaming. According to a later pension filing, he may have married a woman named Georgianna (George Anna) Thompson at Muddy Creek, Saline County, in 1859. According to a family friend named Harrison White, the two were separated when their master, William Piper, sold Georgianna to another owner.

Carr enlisted for a period of three years as a private at the age of 28 on February 1, 1864, in Sedalia, Missouri, and mustered in on February 13 at the Benton Barracks in St Louis. His service record describes him as 5 feet, 6 inches tall, with black hair, black eyes, and black complexion. He first served in Company K of the 67th USCT Infantry Regiment. After leaving Missouri, the regiment first served in the District of Port Hudson, Louisiana, until June when it became part of the District of Morganza. In March and April 1865, Carr performed daily duty in the Post Ordnance Department. Per Special Orders No. 186, Carr transferred on July 12 to Company I of the 92nd USCT Infantry Regiment. At that time, the 92nd USCT was stationed in the Northern District of Louisiana, Department of the Gulf, and saw service near Morganza in September. In late November, Carr was absent sick in the post hospital in Port Hudson, and he mustered out of the service with the rest of his regiment on December 31 in New Orleans.

Sometime after the war, Carr married Silvia (Silva) Gray, and the two lived in Elk Fork in Pettis County, Missouri. According to the 1870 census, Carr worked as a farm laborer and owned land worth $100. They had four children: Susie, born May 3, 1872; Elizabeth; Millie; and Silas. Eli died in 1875 from tuberculosis, a disease that his daughter Susie claimed he had contracted in the army. Silvia died the same year, only six months after her husband. In the 1890s, their daughter Susie Smith tried unsuccessfully to get a pension or any wartime pay in arrears for her father's service. Georgianna, Carr's first wife, who had married again to a man named Patterson, also filed a bounty and arrears claim after Eli's death, claiming the two had a daughter together named Mildrith. Georgianna, however, incorrectly stated that her first husband had died in 1868. Despite producing evidence of their marriage and separation before the war, her application was ultimately unsuccessful.