I need to clean a plain, short pile carpet that is a pale blue colour
and, as a result, shows up the dirt all too well. It has accumulated
various stains over the years and, to be honest, only an area about 4
x 4 feet badly needs a clean.

What's the best way to achieve this CHEAPLY?

I've already tried getting down on my hands and knees with various
'safe' cleaning fluids, trouble is that it's not only hard work but
takes ages.

Is the best option to employ a professional to clean it, or is there a
better and, as mentioned, cheaper option?

Don't some standard vacuum cleaners also clean carpets?

Thanks