The primary tissues in the human body that do not need oxygen are the corneas in the eyes and the outer layers of the skin. A cornea does not have blood vessels to carry oxygen because it if did, the blood vessels would block light from entering and you would not be able to see clearly. So, oxygen from the air is dissolved in tears and then the oxygen is distributed or spread throughout the cornea.
Also our skin is able to absorb a lot of the oxygen it needs directly from the air. According to a study published in The Journal of Physiology, "the upper skin layers to a depth of 0.25-0.40 mm are almost exclusively supplied by external oxygen, whereas the oxygen transport of the blood has a minor influence."