The connection between information entropy and thermodynamics entropy is pretty well established, as well as theories which relate the destruction of information to irreversability. This part of the article is actually not particularly controversial. The idea of a storage device gaining some mass when information in placed on it is not particularly new, but it is something which would be extremely difficult to measure. In terms of the question of absolute zero, you have to consider that the physical reality of a system at zero temperature (truely zero, not femtokelvin) and thus zero entropy is extremely weird: it would basically require a perfect crystal lattice extending infinitely in all directions. This is one of the reasons why it can't be reached. So the answer is basically that if you were to cool a hard disk to absolute zero (which is impossible), it would first require turning it into something not recognisable as a hard disk. (And if you're thinking we've come close to absolute zero in experiments, remember that the difference in scales between thermodynamic information/entropy and other information (even the information contained in the shape of a solid object, let alone any amount of information our current technology could store) is incredibly vast: the temperatures reached in labs don't really come close to that.

So while this is still basically a theoretical idea (and likely will stay that way for some time without a very clever experimental design and a lot of resources: notice they basically propose building a LIGO to perform their experiment), it's not as weird as you might think, and the ways in which it is apparently weird reflect a weirdness that is already present in thermodynamics.