Jensen was born in Herning, Midtjylland, Denmark, on 7 May 1908, as the eldest of five children born to parents Hans Jørgensen Lindholm and Helfrida Nielsine Lindholm. One of her sisters was developmentally disabled, and since the father did not want help from the public, she was responsible for much of her sister’s care. Her father was a blacksmith. In her youth, she would go with her mother to clean three of the city’s offices every morning. After finishing school, she got a job as a telephone operator at Jydsk Telefon, working night shifts.

In 1934, she married Ejner Jensen. They had a stillborn daughter not long after their marriage, and Karla had a number of spontaneous abortions. They later adopted two children in the 1940s: Anne-Lykke (1944–2014) and Hans-Boy (born 1948). Her husband was a master painter, but later established his own business on Herning’s main street. He owned a shop that sold wine, tobacco and handmade pipes. He was in charge of purchasing, while Karla was a clerk and was responsible for the accounts. Her husband passed away in 1975.

In 2014, her daughter died from cancer. Even as a supercentenarian, she still lived in the same apartment where she moved in with her husband following their marriage.