Is converting a BSW to Ethernet jack ends possible?
You can certainly put some ends on the cable and find out what happens. Chances are it'll be CAT3 cable at best and the Ethernet may link at only 100 Mbps or 10 Mbps (if at all). But maybe that's good enough for you.
If it's not good enough you might look at some of the G.hn equipment. The G.hn standard can be used to bridge Ethernet over AC powerline, coax, or twisted pair cable up to 2 Gbps. Most G.hn devices I've noticed support only one or maybe two of these physical media.
Failing that there are of course a multitude of wireless options in 2.4, 5, and 60 GHz unlicensed spectrums, or of course fiber.. But all of that goes well beyond the scope of your question.