The manufacturers of modern sequencers (both Illumina and ABI) have been talking about this for at least 7 years (i.e. as long as they've been selling high throughput sequencers). They actually made a weaker claim: According to them, it makes no sense to keep a sequenced genome, because just sequencing it again would be cheaper than storing the data. In these 7 years, it hasn't happened. Instead, ABI's SOLiD technology all but vanished. Actually storing data in DNA is one step further, it's not going to happen for a long time.
(Source: My employer does a lot of sequencing. I talked to sales representatives of both companies, and I work on data sequenced using Illumina's machines. We store that data on spinning rust.)
And sequencing will become even cheaper when you do not do it from a library prep but in a controlled buffer environment. It is just currently not getting cheaper because there is no incentive for Illumina to do so (similar to Intels position in CPUs), lets hope that ONT, BGI and who ever else still hopes to get some market share (Ion Torrent, PacBio ...) can force them to evolve (project firefly, yeah).
But with a few new players on the block (Twist, Gen9, and a few other smaller/newer startups), the goal is to hit economical ~2-3kb, at which point the race is back on again, and whole new markets will open up. And the moment that happens, expect the price to drop again. Competition will kick back in and everyone's price will drop.
The size of a moderate plasmid (~5-7,000) is another hurdle, and the size of a small chromosome is another (~100,000).
Also, if you're ordering DNA in pools or bulk (have a good compression algorithm), you can get the price/bp to come down even more.
[0] - http://memory-alpha.wikia.com/wiki/The_Chase_(episode)
[1] - https://en.wikipedia.org/wiki/The_Chase_(Star_Trek:_The_Next...
how does this work? are the mistakes consistent enough that we can design encodings that rely upon them?
I don't see why you wouldn't use a higher fidelity atomic storage solution.
Microsoft is already initiated towards this.