Can bad eye sight be improved by eating carrots?

No, but eat them: Antioxidants in the diet are very good for your overall eye health (carrots, leafy greens, blueberries, overall high intake of fruits and veggies). These primarily impact the health of the retina over your lifetime, and may have an effect on the development of cataracts, though this is a weaker association. These good things, though, won't change or decrease your need for glasses or contacts.