Quick search says, a single sperm contains around 787MB of data. The average speed of one of those swimmers is around 5mm/min and on an average there are around 280 million of them in the race.
Curious folks can calculate the bandwidth themselves.
DNA data is interesting but I have to figure that it would be more interesting to encode information into protein. DNA is 4 bases, more information than binary certainly, but with protein chains you could use all the amino acids there are.
Apparently 500 have been found in nature so far although us humans personally use far fewer. It would also seem to me you could engineer a stronger protein structure then how DNA is structured.
Sure, but existing protein sequencing and synthesis technologies are severely limited. You'd lose so many orders of magnitude in scale, speed and cost that the increase in the number of characters wouldn't be worth it.
If you've ever watched the movie Zero Theorem, their data analysts use data captures in fluid dynamics to manipulate and process the data. I always that that was a really interesting concept.
Now if only we could build a molecular Turing machine that can replicate and distribute the data so it is processed in an ever increasing parallel way.
Extensive replication and error correction could go a long way to making this at least remotely viable.
0: https://en.m.wikipedia.org/wiki/Parchive