Sorry. the closest would be Thin Ethernet, but the loss due to a completely different frequency range through coax (I suspect) would be too great for it to be of use. Further, even if you could utilise it, it would be almost impossible to terminate onto a BNC connector. This is now an old system with a maximum theoretical throughput of 10Mbs, whereas the standard is now 100Mbs, with many new networks utilising the new Gigabit standard.
Your best hope I would have thought, is to use that cable to pull a new one; maybe?