Question : Problem: How do you make a CAT5 cable from a 50 Ohm Balun so that the BNC signal gets converted properly?

I have an older system with a BNC / 10BASE2 connector.  I connect to it via my network with an older hub with both RJ45 connectors and one BNC connector.  The hub bit the dust.

I found a 50 Ohm Balun at an electronics store.  Problem is that I get no connectvity when plugging into my 10/100 switch.  I have tried both standard and a crossover cable.  I have made sure to connect the Balun to a BNC 'T' and have a 50 Ohm terminator on it.  I even tried it without the 'T' just for kicks.

The only markings on the Balun are a '+' on pin 4 and a '-' on pin 5.

Answer : Problem: How do you make a CAT5 cable from a 50 Ohm Balun so that the BNC signal gets converted properly?

Not gonna work. The Balun is to convert your line to UTP for the run and then to convert it back at the other end for coax again. You would normall buy 2 of these. One for each end of the run so that you can use UTP instead of coax in the middle. What you need is an active media converter that converts from coax to 10baseT.