Yes but myths abound. Yes, carrots are good for the eyes, but for most. Are no better for the eyes than leafy, green vegetables. I say that because most people in developed nations get more than enough vitamin a in their daily diets than is needed to assure proper photoreceptor function in the retina (what vitamin a is necessary for). So taking more vitamin a doesn't improve vision, but an absolute dearth can hurt.
Yes. Carrots contain vitamin a which is good for the eyes.