If the fate of the universe is a big crunch....

Black holes are pretty much the first thing you’d expect to get out of a theory like General Relativity. The Schwarzschild metric applies outside of any spherically-symmetric mass distribution, and there are a lot of those we’re interested in (you couldn’t really call any theory of gravity complete if it can’t address that situation). And once you have that, it’s only natural to look at the implications of that metric in various situations (like, for instance, that the mass distribution has a small radius). Once you do that, hey, look, black hole.

If a Big Crunch were in our future how small could the universe get before it affects us? From our perspective would it start with the Moon crashing into the Earth, or would it end that way?

It’d end that way, or at least be really, really close to the end.

Big Rips are more fun, though.

It seems obvious today, but Oppenheimer does deserve considerable credit for his insights. The fascinating historical sidebar to this is that at around the time of Oppenheimer’s paper, there was strong opposition to the idea of black holes from scientific greats like Sir Arthur Eddington. When Chandrasekhar presented his paper at a meeting of the Royal Astronomical Society in 1935, Eddington totally demolished him:
On January 11 1935, all the leading figures in astrophysics were at the Society. Chandra delivered his paper, showing a graph that made it transparently clear that a star of above a certain mass would inevitably dwindle to nothing and beyond. Triumphantly he sat down, assuming that Eddington would support his conclusions. But to his horror Eddington, a supercilious man, instead used the full force of his famed oratorical skills to demolish the young man. Had Eddington befriended Chandra in order to destroy him?

Chandra’s theory was mere mathematical game-playing, Eddington argued, with no basis in reality. How could something as huge as a star possibly disappear? Eddington’s arguments were unfounded and highly dubious; but the weight of his reputation was such that no one dared disagree with him. Chandra was not even given the opportunity to reply.

Wikipedia has an interesting note on this incident:
Chandrasekhar’s narrative of this incident, in which his work is harshly rejected, portrays Eddington as rather cruel, dogmatic, and racist. Eddington’s criticism seems to have been based on a suspicion that a purely mathematical derivation from relativity theory was not enough to explain away the seemingly daunting physical paradoxes that were inherent to degenerate stars.
But it also seems that Einstein himself had problems with the idea:
In his 1939 paper Einstein credits his renewed concern about the Schwarzschild radius to discussions with Princeton cosmologist Harold P. Robertson and with his assistant Peter G. Bergmann. It was certainly Einstein’s intention in this paper to kill off the Schwarzschild singularity once and for all. At the end of it he writes, The essential result of this investigation is a clear understanding as to why Schwarzschild singularities do not exist in physical reality. In other words, black holes cannot exist.