Is there a way to get rid of hereditary dark circles under the eyes, without surgery?

Unfortunately, no. These dark circles are caused by fluid accumulation, an increased density of veins, loose skin, excess fat, or a combination of the above conditions. Aside from cosmetics which hide the appearance with the equivalent of a skin "paint", only surgery can improve the dark circles. If yours are caused by allergies, your doctor may prescribe antihistamines to see if they help.