It's like the nostalgia about the "Summer of Love" and the 1960's... it really only lasted a single summer and only in one or two little areas.
Same thing with the "old web." It was about the very early 90s before Netscape Navigator (the Mosaic days) and when everyone was just throwing up a single HTML page with a bunch of links... that's the "old web".
The modern WWW kicked off with the ability to make credit card transactions online (1994). That... and porn (1995).
For "old web" sites that still exist, check out wiby