Not a website, but since you mention BBSes...one thing that would look pretty familiar to a 1990s Internet user would be most of the text-based MUDs, the ancestor of MMORPGs, that are around.

The MUD Connector is still around, and still has a list of active MUDs.

While I suspect that most dedicated MUDders use dedicated clients, the base protocol is still normally telnet, and you can use a plain old telnet client to play...a protocol that predates Internet Protocol itself.