I have heard hassidim cite a source I've since forgotten that it is up to us to bring forth Messianic age through natural means. But if this isn't achieved within a certain span of years (I can't recall the numerical value), Moshiach will arrive through supernatural means. Is there any validity to this? EDIT: I have altered references of HaOlam Haba to allude the different concept of the Messianic age.
The nuts-and-bolts sources of Jewish law sound like the Third Temple will be built by humans using entirely natural means, when it becomes clear that it is the right time to do so. And the prevalent opinion in the Talmud (which is also codified by Maimonides) is that the Messianic era will have the physical world working exactly the way it does now.
There are colorful midrashim, however, that talk about things like the pre-built Temple descending in a ball of fire. One could argue that those are intended metaphorically -- it will happen so suddenly, it's as if it came down from Heaven ready-made. (Similarly the Talmud states that "in the future, fine cakes and linens shall emerge right out of the earth of Israel." One children's publication drew cakes and clothing growing on trees, with the caption "this is what will happen when Mashiach comes." Maimonides in fact gives this specific example in his introduction to chapter Chelek of Sanhedrin -- it's an allegory to describe that the land of Israel would become a place with incredible economic production -- something that I believe we have lived to experience. The Talmud only said "in the future", not "when Mashiach comes.")
But Rabbi Chaim Kanievski and others suggest that there are both natural and supernatural options for the emergence of the Third Temple. Isaiah states that b'itaH achishena, "I shall rush it in its right time", and we are told that means that if we are worthy it will be rushed; if not, it will come in its right time no matter what. Some suggest that if we merit it, it will arrive supernaturally; if not, we'll have to do it ourselves. Others reverse that thinking -- if we merit it, we get to do it ourselves. If instead the clock runs out, then G-d effectively says "I'm doing this for Me even though you don't deserve it", and boom, it miraculously appears.