NOTHING is outside the realm of science. The only thing outside the realm of science would literally be nothing. Something which does not exist.
As awesome for that to be true, its not. Science is horrible at value judgment. For which science can only informed. Science cannot deal with things outside the natural world. It can only examine when these super/para-natural things interact with the natural world, and test them indirectly. Which science doesn't mine doing. If inference didn't work, science would be really fucked.
I'd say that the proper field of science to counter religion with is not math or astronomy, but neurology. Many people who believe in some divine force, do so because they have had a personal experience that -cannot- be shared with others, cannot be reproduced, but that happens often enough that it can be documented. Neurology may find an explanation for this; it's already managed to document some parts. But I still think that the evidence for God (or for whatever) has been very reasonable for a long time.
That evidence is: I experienced something and describe it as X. You experience something, too, and describe it as X as well. So does that guy over there. After a while, these similar descriptions start to form a trend. And after a longer while--let's call it centuries--the dissimilar descriptions also form a trend. It's like a whole lot of blind men--millions--all touching just their own small part of an elephant, and each giving a name to their part of the shape, until the whole becomes describable. They may each be unreliable, but overall, the fuzzy cloud of data points coalesces into something recognizable. Regardless of whether they get accurate results, and regardless of whether there is error in here, this IS science. Religion IS the science of the unknown. And it's fair, in my mind, to call holy books their theses, their collections of data points and interpretations.
Now, modern neurology may be able to come in and say "Your description of what you experienced fits a divine power. It also fits this neurological event that we can produce whenever we want. Therefore, brain weirdness is a valid and more accurate theory". And modern psychology may be able to come in and say "You claim that a thousand people all felt the same thing, but double-blind trials in other fields demonstrate that this proves nothing and is just mob psychology"... and at that point, the onus is on religious people to congregate in smaller groups, try and document their experiences with scientific rigor, stuff like that, if they want to prove their point.
It is very important to note that, among those millions of blind men who are mapping out a shape to discover that it is an elephant, surely many people among them would walk right past that elephant and never find it. With no experiences, surely most of them would think that the rest are making it up. Surely any atheist who never has a religious experience should think that everyone else is crazy, or lying. And surely some of those blind men that missed the elephant are lying about it anyway. I just ask that they consider that some of us may have found what feels like an elephant, even if they haven't stumbled across anything yet. And not all of us even went looking for it. Hell, going looking doesn't mean you'll find it either.
TL;DR: Well of COURSE atheists think religious people are crazy; they haven't experienced the same things!
Well the quick rebuttal is that the plural of anecdotal is not evidence. Anecdotal is the weakest form of evidence, at most it can say there is something to investigate, but it alone can never prove anything.
There variety of reason why anecdotal evidence is near worthless.
No independent verification: There are no data points which aren't anecdotal in origin, or heresy. Anecdotal evidence cannot reinforce anecdotal evidence.
An example would be finger pointing. A and B blame each other for incident C. The evidence that A or/and/nor B is the cause for incident are the anecdotal evidence. Without a third party data point there no way to authenticate the anecdotal evidence of either A or B.
Human Memory is Fallible: Human Memory is great at very certain tasks, such as recognizing and remember faces.
It sucks at remember events, and worse so under stress. These memories accuracy grow worse with time. They are also amendable with little effort.
Peer Pressure: There a sense of needing to conform with groups wants and expectation which may make someone claim an event which did not happen, or embellish details to to conform with group exceptions.
Memory Self Reenforcement: Colloquially known, as believing in your own lie. Amended memories, or fabricated memories become more true to the individual who holds them with time. These memories can become so strong that they can even pass lie detectors. Falsely implanted memories can not be detected with various brian scans, but I do not know if made up memories will also show up.
Human Embellishment: The person reciting the anecdotal evidence may be leaving out all the details of the event, be it accidentally or purposefully. They may also be down playing some events and play up others.
Mass Hysteria: Human in groups can convince a delusion, reenforce itself to be true to the individual, and have quite similar accounts of what happen. Even if the described event can be demonstrated to be false. Mass Hysteria can also be induced artificially.
I'm sure I've missed a few points as to why anecdotal evidence is very weak, but this should be a good overview to why it amounts to nothing.