Can a man take a haircut by a woman Barber?
A man should not take a haircut from a woman barber. Physical contact with the opposite gender is prohibited.
Although the accepted practice is to allow using a doctor of the opposite gender, even when it involves physical contact, see Shach end of Y:D 195. There the assumption is that his involvement in his work and the nature of his work makes the contact purely professional, detached from any affection. Grooming, hair cutting and the like would not enjoy the same exemption and would raise a concern of affectionate touch. Igros Moshe in fact even prohibited a friendly, professional handshake, as even this raises the above concern.