Indeed, the brain is absolutely full of vasculature; I thought this image was a nice example...
(figure from Adams et al 2015)
(a) and (d) are sections through cortex, parallel to the surface. All the black/colored dots are indicating vessels (arterioles and venules) that are diving down into the brain. (b) shows the vasculature of the pial surface, where arteries and veins can be distinguished (color coded in c); this is how they get the color coding in (d), by comparing the overlay position. You can see all the veins, in blue, are larger and surrounded by the arteries, in red.
Note the millimeter scale. The authors report 27.4 vessels per square millimeter.
When describing the blood brain barrier people are not referring to an absence of blood vessels in the brain; rather, the blood brain barrier is between the walls lining all of those vessels and the neurons themselves. Therefore, the actual surface area of this blood brain barrier is absolutely massive. Wong et al estimates 15-25 square meters of surface area over 600 kilometers of blood vessels in the brain, whereas the skin is only about 2 square meters.
(figure from Wong et al 2013)
Elsewhere in the body, nutrients just leak out of capillaries by diffusion. The endothelium in the brain (BMEC in the figure above) strictly controls the transport of most substances from the vascular space to the brain space, and the space between the endothelial cells is tightly closed.
Furthermore, the endothelial cells themselves are covered by the processes of glial (non-neuronal) cells called astrocytes, serving as a second layer of regulation.
Adams, D. L., Piserchia, V., Economides, J. R., & Horton, J. C. (2015). Vascular Supply of the Cerebral Cortex is Specialized for Cell Layers but Not Columns. Cerebral cortex (New York, N.Y. : 1991), 25(10), 3673–3681. doi:10.1093/cercor/bhu221
Wong, A. D., Ye, M., Levy, A. F., Rothstein, J. D., Bergles, D. E., & Searson, P. C. (2013). The blood-brain barrier: an engineering perspective. Frontiers in neuroengineering, 6, 7. doi:10.3389/fneng.2013.00007