They have not yet indicated in the anime or manga exactly what happened at Year 0.
However, please note that the link to the "question close to yours" is almost 6 months outdated, and there is quite a bit of confusion and speculation there which was shown to be erroneous by the manga volumes released since then.
There has been a significant amount of evidence to suggest that Shingeki no Kyojin actually does take place in our universe, in the distant future. Every question of "how could X be like that in our future?" has been explained at this point.
Spoilers from the manga:
The method used to create the walls involved having sentient colossal titans stand shoulder-to-shoulder and encase themselves in a hard crystalline material which they're capable of producing on the outside of their bodies. This explains how they were able to build the walls fast enough to protect themselves from the titan outbreak, and offers one of several explanations as to why much of the technology in their day-to-day lives seems medieval (never mind the certainty of government involvement). The world seen inside the walls is not in any way representative of the outside world which existed before that, since the government has outlawed all information regarding the outside world and its history.
It has also been demonstrated that the titans are all transformed human beings, with one titan being seen later appearing to be a transformed monkey (potentially one of the first experimental subjects). It's also been indicated that Eren's ability to transform into a titan at will was catalyzed by an injection of some chemical from his father. These are a few of the things which show that the titans are almost certainly a result of our own creation, developed through some advanced future technology.
What very few historical facts have managed to be preserved (through the rare informative book which has been successfully hidden from the government) match up perfectly with our own history. Their legends are our legends, their history is our history, and their world is our world.
As for when this takes place, the earliest it could hypothetically be (at the time Eren is in training) is 2116 AD, since the manga was released in 2009, and it was stated when Eren was in training that the titans were created 107 years earlier. This naturally assumes that the government hasn't used their "world history blackout" to fabricate the details of the titan's appearance.
(Writing it out, I just noticed this setting has a few similarities to The Village.)
Continuing that line of thought, where we assume the new government didn't arbitrarily change the dating system just because they could, the "Year 850" would mean that we began using a new dating system at some point in the future, starting at 0 or 1. Maybe another World War? (that's just pure speculation, but it's yet another of the many possible explanations as to why the technology of the common peasants is so underwhelming).
Since that would make this at least 850 years from some unknown point in the future, and "the future" is any time after the manga began releasing in 2009, that would mean the earliest that Year 850 could be is 2859 AD.
However, one last thing to consider is that the first chapter of the manga is titled "To you, 2000 years from now." This shows that a 2000 year gap of time is relevant to the storyline somehow. There are a few interpretations about what this could be referencing, but one of the more popular ones is that this chapter title is addressed to the reader, meaning the storyline starts 2000 years from now.
That would put the beginning of the series in the year 4009 AD. It also means that the new world's dating system would begin in 3164 AD.
Considering the advanced level of technology it's suggested was used to engineer the titans, this appears to have a significant chance of being correct.
It would also give the author an infinite number of ways to explain every aspect of the setting, since literally anything could happen over 2000 years. Especially when something so significant happened halfway through that they felt the need to reset the calendar back to Year 0.
At any rate, I hope this up-to-date information was useful to you.