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.