Can you dissect a square into similar rectangles with aspect ratio 1:sqrt(2)?
I have a suspicion you can't and that a proof could be constructed whereby you make one side of the square an integer length, then prove that the other side has to be non-integral, ie some multiple of sqrt(2).
An 'almost' answer, one side out by about 0.021%. This arose when tiling with similar but distinct rectangles of square or twice a square area.
Rectangles labelled with their area.
Your intuition is right; it cannot be done.
The problem is the same as asking whether a $1:\sqrt{2}$ rectangle can be dissected into squares.
(To see this, note that as pointed out in the comments, any dissection of a square into the required rectangles can be made into a dissection into rectangles of vertical orientation only by splitting horizontal rectangles into two. Such a dissection can now be squished vertically by a factor of $\sqrt{2}$ so that all the rectangles are squares.)
Dehn has shown that if a rectangle is dissected into a finite number of squares, the aspect ratio between sides of the rectangle must be rational, so a $1:\sqrt{2}$ cannot be dissected into squares.
There are some historic notes on this theorem and a proof in this paper: Tiling a square with similar triangles, which goes into a more general version of your problem.