Barbauld was born on 20 June 1743 at Kibworth Harcourt in Leicestershire to Jane and John Aikin. She was named after her maternal grandmother and referred to as "Nancy" (an 18th-century nickname for Anna). She was baptised by her mother's brother, John Jennings, in Huntingdonshire two weeks after her birth. Barbauld's father was headmaster of the Dissenting academy in Kibworth Harcourt and minister at a nearby Presbyterian church. She spent her childhood in what Barbauld scholar William McCarthy describes as "one of the best houses in Kibworth and in the very middle of the village square"; she was much in the public eye, as the house was also a boys' school. The family had a comfortable standard of living. McCarthy suggests they may have ranked with large freeholders, well-to-do tradesmen, and manufacturers. At his death in 1780, Barbauld's father's estate was valued at more than £2,500.
Barbauld commented to her husband in 1773 that "For the early part of my life I conversed little with my own Sex. In the Village where I was, there was none to converse with." Barbauld was surrounded by boys as a child and adopted their high spirits. Her mother attempted to quash these, which would have been viewed as unseemly in a woman; according to Lucy Aikin's memoir, what resulted was "a double portion of bashfulness and maidenly reserve" in Barbauld's character. Barbauld was never quite comfortable with her identity as a woman and always believed that she failed to live up to the ideal of womanhood; much of her writing would center around issues central to women and her "outsider" perspective allowed her to question many of the traditional assumptions about femininity during the 18th century.
Barbauld demanded that her father teach her the classics and after much pestering, he did. Thus she had the opportunity to learn Latin, Greek, French, Italian, and many other subjects generally deemed unsuitable for women at the time. Barbauld's penchant for study worried her mother, who expected her to end up a spinster because of her intellectualism; the two were never as close as Barbauld and her father. Yet Barbauld's mother was proud of her accomplishments and in later years wrote of her daughter: "I once indeed knew a little girl who was as eager to learn as her instructors could be to teach her, and who at two years old could read sentences and little stories in her wise book, roundly, without spelling; and in half a year more could read as well as most women; but I never knew such another, and I believe never shall."
Barbauld's brother, John Aikin, described their father as "the best parent, the wisest counsellor, the most affectionate friend, every thing that could command love and veneration". Barbauld's father prompted many such tributes, although Lucy Aikin described him as excessively modest and reserved. Barbauld developed a strong bond with her brother during childhood, standing in as a mother figure to him; they eventually became literary partners. In 1817, Joanna Baillie commented of their relationship "How few brothers and sisters have been to one another what they have been through so long a course of years!"
In 1758, the family moved to Warrington Academy, in Warrington, where Barbauld's father had been offered a teaching position. It drew many luminaries of the day, such as the natural philosopher and Unitarian theologian Joseph Priestley, and came to be known as "the Athens of the North" for its stimulating intellectual atmosphere. One other luminary may have been the French revolutionary Jean-Paul Marat; school records suggest he was a "French master" there in the 1770s. He may also have been a suitor to Barbauld; he allegedly wrote to John Aikin declaring his intention to become an English citizen and to marry her. Archibald Hamilton Rowan also fell in love with Barbauld and described her as, "possessed of great beauty, distinct traces of which she retained to the latest of her life. Her person was slender, her complexion exquisitely fair with the bloom of perfect health; her features regular and elegant, and her dark blue eyes beamed with the light of wit and fancy." Despite her mother's anxiety, Barbauld received many offers of marriage around this time—all of which she declined.