Why are nitrogenous bases of DNA hydrophobic if they can hydrogen bond? Why are nitrogenous bases of DNA hydrophobic if they can hydrogen bond? Is it that they are only relatively hydrophobic? This forum explains it but does not give an example of the structure.

These two concepts are not mutually exclusive. You have constructed a false dichotomy. Both of these facts are true:

1. The electrons in the pi orbitals of the conjugated double bonds in the planar rings are hydrophobic and can "stack" on each other as shown in this drawing of a DNA helix where the bases are shown as planar rectangles:



!A model of a kinked DNA helix

2. The nitrogen and oxygen molecules (either in the rings, or as substituents on the rings) can participate in hydrogen bonds (H-bonds):



!H-bonds in Watson Crick base pairs

There are other examples of hydrophobicity and H-bonds co-existing. For example on the interior of proteins.