It's a bit too simple to say that that's the problem, it's not that general relativity can't deal with global features of reality, those will simply not look the same to all observers. The reason I mentioned CMB is that space necessarily seems to have a preferred speed (even with general relativity you can't have a distribution of speeds that looks the same from all frames of references), and measuring the blueshift in CMB allows you to measure an absolute speed in some sense (it might still be local to our part of the universe, but that includes pretty much everything we can interact with).

The obstacle to combining general relativity and quantum mechanics is, in short, that general relativity is a classical theory of physics (e.g. exact positions, energy, momentum and all that), whereas quantum mechanics expands classical mechanics to get quantum mechanical laws of physics. Now for whatever reason the techniques we used to turn electromagnetism etc. quantum mechanical fail to work on the (classical) theory of general relativity.

And IMHO that's about as far as we've gotten, a lot of work's gone into it but it's honestly hard to tell if we've gotten a better grasp on why general relativity refuses to 'quantize'. Personally I blame the fact that the mathematical foundations of quantum mechanics aren't strong enough to support a quantum mechanical description of geometry itself, but I may not be the most qualified person to judge this.