I think it is pretty normal and as long as it isn't extremely red, or painful, you should be ok. The whites of your eyes veiny perhaps because lack of oxygen to your eyes. If you are a contact wearer, Basically the contact lenses are cutting off the oxygen to your eyes. the best fix for this is to buy some rewetting drops and put them in every couple hours regardless. If you don't wear contacts then I have no idea. Definitely see a specialist regarding this.
Generally speaking, our eyes would be a little bit clean and pure. Once you got some red or veiny things, it is likely that you have got red eyes due to some reasons such as lack of sleep, bad use of eyes, exposure to strong lights or screen, or other infections. Thus, you should try to consult a doctor and look for some good ways to improve your health conditions.
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.
Everybody has veins in their sclera. And it will be more prominent if you have some eye problem, such as eye infection and eye fatigue. And those veins will be disappear when the eye problems been cured. And you should see a doctor if the prominent vein in your sclera lasted for a long time.
Each person will have different blood vessels in the conjunctive, so some variation is normal. If your eyes are chronically red, however, it may be a sign of ocular inflammation, dry eye or other causes and may be treatable with eye drops. I'd suggest anyone with chronic red eyes to be examined.