Why are RF components and cables still so large?
Why can't coax, as you see on the side of the this amplifier, be
  decreased in dimensions?


It's all down to characteristic impedance of cable: -



If you plug in the numbers, to obtain a centre conductor thickness (d) that is not unfeasibly small, dimension D cannot be to low. For instance if d = 1mm then for a relative permeability of 2.2, D has to be about 3.4 mm to obtain a 50 ohm characteristic impedance. Then on top of this is the thickness of the screen and the plastic outer covering.

These numbers scale down ratiometrically but imagine having a centre conductor of 0.1mm - how reliable will this be and just how much current could it carry?

For 75 ohm systems and a 1mm centre conductor, dimension D needs to be 6.5mm (relative permeability of 2.2).

Characteristic impedance is important just in case you weren't aware of this.