In the old days, computers had real serial ports, and used a serial modem to get online, by dialing to some BBS over the (fixed line) phone network. Today most people have mobile phones, and real serial modems doesn't make much sense anymore, unless you are going to set up your own phone network.
So how can you get a vintage computer online today? One way is via TCPser, the serial to IP modem emulation program. Check out Putting Your Retro Computer On the Line, and find the updated tcpser for on Github: FozzTexx/tcpser. Not bad use of a Raspberry Pi. TCPser is also avilable in FreeBSD ports: comms/tcpser.
Another way is with Hayesduino, source also on GitHub: plbyrd/hayesduino which requires and Arduino with Ethernet shield.