Wow, that's a tough question. Let me give you some info to get you started. First, the cartoon drawings found in textbooks of phospholipid bilayers are inherently going to misrepresent the complexity of the system. Membranes are not just "bilayers." In fact the term bilayer comes from the fact that the membrane can be split it two sheets when examined using freeze-fracture techniques (the same technique used to first study biological membranes). It is more accurately thought of as a semi-liquid/semi-solid multi component layer.
Anyway, to answer your question these fatty acid tails don't just point at each other in straight lines. They bend and bind, and hold protein transport molecules in place. The lipid bilayer does have a hydrophobic core. In this core, you have interactions between the hydrophobic tails that cause them to chemically bind (ionic bonds if I remember correctly). Their ability to bind effects how leaky the membranes are. How? Well, I'll let you figure that out. Here's a factiod that may help you think about this question. Deep water fish that live in cold water at high pressures have membranes with very few saturated bonds (in other words, their fatty acids are unsaturated), while cows that live at warmer temperatures and atmospheric pressure have higher saturated fatty acids. Think about why this is. It has to do with the concept of homeoviscosity. Find that word, and you'll find the clues that will help you answer the above question.
Welcome to the wonderful world of physiology!
good luck and don't give up. Write back if you've got more questions.

P.S. Check out the website:
http://www.biosci.uga.edu/almanac/archive/spring_97/bio_103/notes/may_15.html[]

Under most circumstances, x-ray diffraction shows that the tails of the phospholipids that make up cell membranes align end to end and do not overlap significantly.This is to maintain a uniform density within the bilayer similar to the density of an alkane. Problems arise when the packing of the polar parts, or headgroups, dictates an area per molecule larger than can be accommodated by the tailgroups. The tails handle this by tilting - essentially the bilayer gets thinner. However, if tilting can't get the job done, the chains can interdigitate - that is overlap in the middle of the membrane. The most common way for this to happen is in the presence of ethanol. The ethanol gets into the headgroup region and pushes the heads further apart, so far apart that the tails have so much room that they can interdigitate.

It is possible for lipid tails to interpenetrate in some cases but they apparently do not do so in cell membranes. We know how thick cell membrane lipid bilayers are from microscopy and radiation scattering techniques. The thicknesses are consistent with the non-overlapping images depicted in textbooks.