I remember the good old days of dialing into a BBS before the Internet was really big. You could use a terminal emulator included with Windows or something more specialized like Minicom in DOS. Sometimes you'd have to wait for hours to get connected because someone was tying up the telephone line and you'd end up getting busy signals. Then finally, you'd get connected! Then you could login and navigate pretty much a text-only system, with the only graphics being ASCII art or anything you downloaded and viewed offline. I had a 2400 baud modem through most of my BBS days. I think it took an hour to download 1MB. Anyone could run a BBS, it just took a dial-up modem and software on your computer.
What would a modern one look like?
I think it would be based entirely off of the Internet technologies that have become more popular since the 90's. You could set up a dial-up server with TCP/IP connections. Anyone who connects in would get a private IP address, then have a captive portal (such as used in wifi hotspots) to redirect people into a webserver you have set up. You could run a phpBB instance on the webserver and call it a day. That would get you forums, you could run other servers for other services. For example, have a FTP server set up for files. The users might even upload a file and link it in a forum post. Or run an email server for the users.
Depending on location or access to locations, it could be possible to set up wifi access points to allow for high-speed access to this modern BBS. It's a shame that DSL isn't a little more like dial-up, not speed-wise but with dial-up's ability to just call anyone and get a digital conversation going.
Uses? Community-oriented network, really private network, backup communications path if a group loses access to the Internet, maybe useful in places like Egypt etc. I could see a case where wifi access is available and someone walks by with a Windows CE palmtop or similar device to send/receive email real quick.