Can DNA base pairs go through each other?
The base pair you are describing is only stable if each base is part of a macromolecular polymer, a polynucleotide. Specifically the DNA strands need to be oriented in an anti parallel alignment, and there will be a chain, or string, of base pairs (imagine the rungs of a ladder).
Remember, individual hydrogen, or H-bonds, are one of the weakest molecular interactions that we can detect. So the scenario you propose, where one single base forms a Watson-Crick base pair with its complementary base, free in solution, is so unstable as to be undetectable.

Once the base pairs are formed in a DNA double helix (think of a flexible ladder where one end is fixed and the other end has been twisted through several clockwise revolutions) then there are many other energetic contributions that stabilize the base pairs. For example, the electrostatic repulsion of the anionic phosphates in the sugar-phosphate backbone drives the two strands apart.  The hydrophobic stacking interactions of the pi bonds of adjacent bases also makes a major contribution to the stability of the helix, and permits the H-bonds to be more stable in this context.

Finally, the fact that base pairs are only stable in polynucleotides means that if you melt a single base pair, the sugar phosphate backbones will topologically prevent any other melted base pair from "passing through" the first one.