A lot of mathematicians I've known feel a kind of awe for Hilbert that they don't feel even for history-makers like Euler or Gauss. I think little stories like this one are part of the reason why. Hilbert always makes it look easy, even when his results are weirdly counter-intuitive.
Technically, though, there aren't any paradoxes here. It's for the purpose of avoiding paradox that we're willing to talk about infinities in the first place.
Technically, though, there aren't any paradoxes here.
The real reason why it is a paradox doesn't have much to do with notion of infinity per se, but with problem of infinite decision aka Supertask.
Which reminds me of an article on physics of supertasks that among other things somewhat outlines how classical theory is more paradoxical before than after quantization http://th-www.if.uj.edu.pl/acta/vol36/pdf/v36p2887.pdf (43pp but easy reading).
Technically, though, there aren't any paradoxes here. It's for the purpose of avoiding paradox that we're willing to talk about infinities in the first place.