It is not normal to have veins in the eyes. The red veins are blood vessels in your eyeballs that is are irritated, dilated, and swelling. The irritation or dilation is caused by several major factors as followings: dry air, exposure to sun and dust particles, foreign body in the eye, fatigue, Blepharitis, pink eye, corneal ulcers, Uveitis, acute glaucoma and scratching in the cornea. To have your veins removed, you should have a good rest such as a good sleep in the night. Cold or warm compress is applicable for treating the veins according the cause of the vein. You should notice that you cannot wear contact lenses if you have veiny eyes. I also suggest you to turn to doctor for timely treatment.