If you have a piece of proper music,
eg https://www.youtube.com/watch?v=nJSyAjx2rHA
If there any way in Audacity to make it retro-8-bit sounding (or even 16 bit) like:
Short answer: No.
Longer answer:
The reason that the second tune sounds “retro” has nothing to do with “8 or 16 bits”. It sounds retro because of the simple synthesizer sounds that it is being played with. If you play any tune on a late '70’s / early '80’s home computer synth, it will sound “retro”. If you play the same tune on a piano or a trombone, it won’t sound retro.
If you are interested in creating that style of vintage computer music, search Google for “chiptunes” or “make chiptune” or similar. There’s a niche scene of chiptune enthusiasts and software available for making it.
(It is “possible” to create that style of electronic music with Audacity, but it’s difficult and it takes an enormous amount of time. There are much better programs for creating chiptunes.)