After hearing the recent announcement that scientists have stored a movie in DNA, I am reminded of a short story I read about a very similar concept; but I can't remember where I read it, the author, or its title. I only know that I would have read the story in the late 90s or early 2000s and it's possible that the story is decades older than that.
A company sold a drone that followed you around and acted as your personal videographer. The company would convert the movies it recorded into a DNA sequence and store them in a vat. The idea was that as a viewer you weren't able to choose which video you watched, but this was by design so that the videos could not be subpoenaed in the event that you did something illegal (I think?).
The story was told from the point of view of a deceased man's family who would occasionally (often?) view his recorded memories only to find that they kept seeing the same small set of movies, and over time the DNA storage degraded, leaving them with increasingly distorted video until it was all but indistinguishable. I remember the the tone of the story being somewhat melancholy or bittersweet.
Edit to be more specific about the most likely collections where I may have read the story (I have read many more, this list is not exhaustive):
I think I read the story in high school (class of 2000); our library had a small collection of short sci-fi compilations. The compilations were from mostly golden age and new age authors (50s-70s) such as Asimov, LeGuin, & Ellison.
My second guess would be that I read it in one of the "Year's Best SF" collections by David G. Hartwell & Kathryn Cramer. I no longer own these so I can't skim the stories.
Lastly (a long shot because I'm pretty sure the story DIDN'T win a Hugo) but I own a two-volume collection of the first 10-20 years of Hugo winning short stories, edited by Asimov. I'll look through these tonight and see if I can find the story.