First a look at the adenine-thymine base pair. Thymine is on the strand whose 5' phosphate group is shown on top. The strand with adenine goes the opposite way and you see an open 3' oxygen. There are two hydrogen bonds in the AT pair.
Let's look at this base pair from the other side. The important point is that the backbone looks the same as in the above image. And ths is why you can have alternating A's and T's in a strand. Apparently this is what Francis Crick pointed out to James Watson, when Watson had suggested base pairing. Considering that this had to be pointed out even to the person winning the Nobel prize for discovering the DNA structure, I think it is safe to say that people without this understanding of helical symmetry do not have to feel bad about it. But apparently it is a non-trivial fact.
The next non-trivial fact is that the backbone of the guanine-cytosine pair looks just like the AT and TA pairs. Here you can see the base pair with its three hydrogen bonds and which is thereore bonded a little bit more strongly.
And again, CG also fits.
It seems therefore that one reason for the specific choice of bases is that they all fit into the same helical structure. And therefore it is possible to have a stable helix which contains the genetic code through alternating bases. Another source of selection pressure was probably photostability, as explained here.