Those are just subsets of R though (though I think your last example would take more work to make rigorous, if it's even possible).

The weirdness of the axiom of choice only really comes through when you consider bigger and bigger collections of sets.

For example:

- sets indexed by the natural numbers: S_1, S_2, ... It seems totally reasonable that you should be able to make a new set by picking something from the first set, something from the second set, etc

- sets indexed by continuous time (i.e. real numbers). Here it's a bit less 'obvious'. If I have sets S_t for _every_ time t > 0, can I really make choices 'fast' enough? What if the sets are so unstructured that I'm forced to stop and look at each set in turn to make my choice?

- sets indexed by the power set of the real numbers. If you weren't convinced that I'd struggle to pick elements of S_t for all t > 0, what if I had to make a choice for every _possible combination_ of real numbers, infinite or otherwise?

I feel like the last example demonstrates how powerful the full axiom of choice actually is.

NB - I'm a dilettante rather than an actual logician, so there may be mathematical inaccuracies here.