Marius Diepenhorst has pioneered the following technique.  He writes (2004):

"Try to get a LANTRONIX UDS-10 device.  It acts like a modem but it is a LAN
-> RS-232 converter.  So with that device you can have incoming and outgoing
'calls' like modem ones via the internet.

I ran my Atari 8bit bbs with such a thing. The Lantronix MSS-10 or MSS-100
will do too.  But in that case you have to make a custom RS-232 cable (easy
job).

More info www.lantronix.com

this is the info of the UDS-10

www.lantronix.com/device-networking/external-device-servers/uds-10.html
  Now see the newer model, the UDS1100:
  www.lantronix.com/device-networking/external-device-servers/uds1100.html

It is REALLY a cool thing. Not only for you, but for more atari fans I guess."

Other, similar serial-to-Ethernet interfaces from Lantronix have been
successfully utilized, including the MSS100:
www.lantronix.com/device-networking/external-device-servers/mss100.html

as well as the discontinued MSS1-T.

Note that the UDS-10 lacks DNS support, while the MSS100 and MSS1-T include
DNS support.