yeah get is use in this case, or I need a hair cut. I need to get a haircut,
don't use "an" with consonants or with consonant sounds ex. a book, a pencil, a house is well spoken.
an apple, an hour because this is a vowel sound.
It would seem to me that if you step into a barbershop it is obvious that you want your hair cut. You would need to have a few more words to use. Usually you tell the barber how you want your hair cut. Do you want a full haircut or just a trim? None off the top, just around the sides, etc. A buzz cut or razor cut, whatever. You are leaving it to the whim of the barber by just saying, "I would like a haircut, please".
Jacinta, your post is exactly what I was thinking! If I were a barber, and someone asked if he could get in a haircut (in a barbershop, mind you), I'd be tempted to say, "No, we only sell flowers here," or something similarly nonsensical!