Unprovability of GCH is blowing my mind again

When I first learned that it can't be proven to be true or false I was taken aback. Like how is that possible? But learning about it, realizing it actually just means ZFC doesn't disallow either assumption and you can either take ZFC+GCH or ZFC-GCH calmed me down a bit. Okay, differently constructed axioms, different results, that's fine. I can think of it kind of like whether I allow square root of -1 determines whether I have C or am restricted to R. It's still a very interesting observation, but fundamentals of my understanding aren't crumbling under my feet.

But suppose you take ZFC-GCH. Does that mean there exists a set that you can't biject with integers because it's too big, and you can't biject it with all reals because it's too small? With my very limited understanding of GCH, I don't think -GCH necessarily forces this particular cardinality to exist, but the fact remains that it implies existence of something like that: a cardinality strictly between aleph_n and aleph_n+1. Now maybe if I understood cardinalities beyond that of the reals, this fact would be less of a surprise, but then again maybe not. For now trying to think about any such cardinality I have to rely on my understanding of these two that I do understand. So suppose there is a subset of the reals that's bigger than the integers and yet can't be bijected with all reals. What could it possibly be? I mean, the existence of it implies you can describe it, right? At least you should be able to provide a number of examples and counterexamples of members of that set.

But ZFC+GCH is just as correct. So what happens to this set then? Intuition says it would become bijectable with the reals / the higher aleph, which might be incorrect, but more importantly: the bijection itself! It becomes illegal somehow. Which makes me think GCH, despite being conceived to answer questions about set cardinalities might involve a lot more.

I don't know why it suddenly started to bother me. The thought came to me and is threatening my understanding of math. That I don't understand something like the complex arcsin doesn't bother me much, that is in some sense too fancy and yet clearly mundane. Understanding how stuff fits into boxes feels something one should understand instinctively, and when you're confused about it, it feels like some fundamental part of your mind is threatened. First time learning about countable vs uncountable or various infinity paradoxes is something we can probably all remember being fascinated by. You can probably remember questioning things yourself and eventually coming to understand them and looking down on the morons who don't. So whether this reads like gradeschool homework to you, or you are wondering the same thing, or even if you're like "what's GCH?", I think you can sympathize with my perplexedness and my quest for demystification here. Oh, and then there's the "there's more infinities than there's anything else" thing. My brain is hungry for answers... dumbed down to where I can understand, and I'm not sure if they'll ever be