Does a woman have to have a mahram with her to leave the house, and what is the evidence for this from the Qur'an or Sunnah?

There is no evidence that obliges a woman to have a mahram with her when she leaves her home unless she is traveling, based on the Prophet's (peace and blessings be upon him) saying: "A woman should not travel except with a mahram." A woman is permitted to go out alone for her needs if the path is safe and her outing is not considered travel. Indeed, the women of the Companions used to go to the mosque and seek rulings from the Prophet (peace and blessings be upon him) alone. The default is for a woman to remain in her home, as Allah Almighty says: "And abide in your homes." However, she is permitted to go out for a need, based on the Prophet's (peace and blessings be upon him) saying: "Allah has permitted you to go out for your needs."