Science calls the hair on our head the hair shaft. Cells made of a protein called keratin form this shaft. These cells become fully keratinized and lose their life functions by the time they push through the skin. That is why the visible hair has no blood vessels.
But the hair roots inside the skin contain living cells. Capillaries surround the dermal papilla at the base of the hair bulb. These blood vessels provide the oxygen and nutrients needed for hair growth.
Similarly, we call the visible part of the nail the nail plate. Hard keratinized cells make up this plate too. These cells lose their nucleus and harden as they mature. So, the visible nail plate lacks blood vessels.
However, the nail bed underneath and the nail matrix contain living tissues. Blood flows actively to these areas. A wide network of blood vessels in the nail unit supplies oxygen and nutrients to the growing parts.
So, saying hair and nails have zero blood supply is not entirely true. The visible hair shaft and nail plate lack blood vessels. But the hair follicles, nail matrix, and nail bed get plenty of blood. That is why cutting your hair or nail tips does not hurt or bleed. But injuring the hair root or the nail bed will definitely cause pain and bleeding.