NO chance. NO one has ever contracted HIV in a barber shop, unless they happened to be having unprotected vaginal or anal sex in the barber shop, or unless they decided to inject drugs and share intravenous needles while getting a shave and a haircut.
HIV is NOT transmitted by any tools or products that a barber would use. Not razors, not styptic - nothing.
Do not have unprotected sex with someone who's status you don't know, and don't share IV needles with drug users, and you'll never have to worry about HIV.