Jeppe wrote:
"Of course the height of the waveguide has to be more than lambda / 4 in
order for the lambda / 4 rod to fit in. But how much bigger should it
be?"

Not critical.

Circular waveguides work but have a disadvantage of allowing multiple
modes. Square waveguides share this disadvantage.

To restrict formation of multiple modes of propagation, elliptical and
rectangular waveguides are used.

We are familiar with the illustrative and instructive diagram of
formation of a rectangular waveguide from countless 1/4-wave
short-circuit stubs attached to a parallel-wire transmission line. That
would require one dimension of a waveguide to be at least
1/2-wavelength. That computes and it has less attenuation if the wave
has just a tiny bit more elbow room.

There is no requirement on how small the spacing of a transmission line
can be as long as it doesn`t short where you don`t want it to short.
There is no abrupt minimum spacing for the small dimension of a
waveguide either. As in the longer dimension of the waveguide
cross-section, the attenuation rises as the guide narrows, but it does
not cut off propagation down the guide at 1/4 wavelength as the longer
dimension does at 1/2-wavelength.

I don`t have a catalog at hand but it seems to me that rectangular
guides are about twice as wide as they are tall. This ratio has proved
to work very well.

See page 265 of "Transmission Lines, Antennas, and Wave Guides" by King,
Mimno, and Wing for confirmation and more details.

Best regards, Richard Harrison, KB5WZI