Our blood is not blue. It is always red. Even when it's deoxygenated. Even in the absence of oxygen in a vacuum. (Remember, when you get blood drawn at your doctor's office, they use a vacutainer, which is essentially a vacuum in a tube. The tube is attached to the needle in your arm, exposing the inside of the vein to the vacuum and drawing the blood out.)
After your blood is pumped to your lungs by your heart, it's bright red because haemoglobin -- the iron-containing, oxygen-transporting protein in our red blood cells -- binds to the oxygen the blood just picked up. From the lungs, the blood goes back to the heart (this is called pulmonary circulation), which pumps it out to the rest of the body via the arteries and into tiny blood vessels called capillaries, where it gives its oxygen to the body's tissues (systemic circulation). On its return trip to the heart through the veins, the oxygen-depleted blood is dark red or maroon, because the haemoglobin is no longer bound to oxygen.
WHY SO BLUE?
Doctors will tell you that when you poke around inside a human being and see a vein or artery in its naked glory, it isn't blue. If blood isn't blue, and veins and arteries aren't blue, why do our veins look blue through our skin?
When you look down at the veins in your arm, the light of different wavelengths is hitting the skin, the veins and the blood. Some of that light is being absorbed, and some is getting scattered and reflected to your eye. Different wavelengths of light have different properties and abilities. As it turns out, blue light, compared to red light (i) doesn't penetrate the skin as well, (ii) is absorbed by the blood more, and (iii) is more likely to be scattered and make it back to your eye.
So, if a vein is close to the surface of the skin, most of the blue light will be absorbed, and even though red light doesn't reflect as much, the red light:blue light ratio is high enough to make the vein appear red. With deeper veins, the blood doesn't absorb as much blue or red light. But the blue light's inability to penetrate as deeply as red light makes the veins appear blue.