We were taught in basic biochemistry that this question was answered in 1965 with a resounding, table pounding "No." Case closed. But on reflection, and at this remove in time, it is clear that the (rather appealing, really) possibility that nucleic acids constitute a long term memory storage medium was neither tested nor refuted 35 years ago. No one knew how. No one would know how today.
The author, who became deeply skeptical of the original 1960s research that launched the idea of nucleic acid memory -- tells the story of this forgotten controversy from a personal point of view, and this is the most interesting part of the book.
The fact is, biological information is very typically stored as sequences and shapes, and there is no reason to imagine the human memory is stored in some entirely different way. Probably the notion of nucleic acid memory will get a second hearing someday when we have the technology to actually test it, and some sort of a hunch or clue about how such a thing might work in the brain.
A fun book on the subject is the science fiction classic, "Hauser's Memory," and it is probably the only other book on nucleic acid memory that is still available.
For a quick, seamless review of the currently accepted view of human memory, which is grounded on the assumption that memory is stored as synaptic changes, see Kandel & Squire's book, "Memory. From Mind to Molecules."
For a sense of why the cherished assumption of synaptic memory will probably fail, and pretty soon, see the recent, mildly written but revolutionary book: "Spikes," by Rieke et al.