How does blood reach skin cells and other closely packed cells?
The skin is highly vascularized up to the epidermis, with dense vessels right at the top of the dermis (your picture omits these, but you will find them in better diagrams like this one). The epidermis itself gets substantial oxygenation from the outside air (see Stücker et al, for example), though the very outermost layers are not metabolically active (&quot;dead&quot;) and do not require oxygen.
As a gas, oxygen moves quite well through tissue, and cells can get sufficient oxygen from sources up to a few hundred micrometers away. No need to be in direct contact with blood cells (and typically no tissues of the body are in direct contact, besides vascular endothelial cells that line all blood vessels). Many small animals have no need for a circulatory system because their entire body is close enough to the outside for gas diffusion.

Stücker, M., Struk, A., Altmeyer, P., Herde, M., Baumgärtl, H., &amp; Lübbers, D. W. (2002). The cutaneous uptake of atmospheric oxygen contributes significantly to the oxygen supply of human dermis and epidermis. The Journal of physiology, 538(3), 985-994.