All this is made possible by the use of so-called "next-generation" or high-throughput sequencing, which is exciting as we here at the University of Birmingham will be getting our HTS instrument soon (in fact, several instruments if the grant proposal I sweated blood to get in earlier this week is successful).
It is a little frustrating that the authors don't provide chapter and verse on how many kits they used and how many sequencing runs they needed to do to get the ~1.1 million reads. But a quick back of envelope calculation suggests that one could replicate their study for around £20-30K today, which is astonishing. They point out in the paper that a full nuclear genome of the thylacine could be achieved for ~$1m, so I guess it is inevitable we will see one within a few more years.
Sadly, it is very unlikely, despite a few die-hard believers and supposed sightings, that any of us will ever get to see a living thylacine. But if you want a good book to read on the subject, try the excellent Carnivorous Nights by Margaret Mittelbach and Michael Crewdson.