What's the diameter of the tube where the protons fly around? I dont mean the overall diameter (~1-10 m), instead the inner most ultra-high vacuum tube where the protons are. The inner pipe, the capillary, the channel.. I would guess maybe 1 mm?
I'm not into particle physics myself but I found this: The LHC: a look inside.
It says there:
Inside its two beam pipes, each 6.3 cm in diameter, proton (or heavy ion) beams travel in opposite directions (one direction in each pipe) in an ultra-high vacuum of 10-13 bar, [...]
The answer is not simple, though the diameter of the beam tubes is given by the link of Jonas's answer. The size of the beam itself is controlled by a series of magnets, and at certain locations the beam is collimated. In page 4 of this talk the emittence in the physics regions (squeezed) comes down to order of μm.So your mm guess must not be too far from the real beam size.
The thing is that a larger vacuum is needed in order to tune the beams efficiently, than the final size of the beams.
In addition technological problems in mm dimensions of metal etc parts will be more challenging and expensive then the extra space of vacuum in the tubes.
According to this article the external diameter of the beam tube is 53 mm, and the wall thickness is 1.5 mm, so the internal diameter is 50 mm. Since this is a CERN server I assume it's trustworthy.