Fencing I have is called dog pen fence wire..it is 48" high and has about the same sized squares as you mentioned Patti. I would think what you have would work, the only thing that you might want to keep in mind, is that the vines may climb over faster and you should leave enough ground space on the other side of the wire for the vines once they reach the ground. I did not have to have to train mine, but it doesn't hurt when they start growing
to place the vines next to the fence wire, they will climb on there own...And Yes it take's alot less space then growing
them on the ground (mine would have taken the space from the wire to where the okra is and probable tried to climb the okra stocks..far left of picture, so I did gain space and saved on my back bending over looking for cucumbers also), plus it seems my harvest was better growing
them up on the fence, there were less missed cucumbers also when I picked them...also you want to pay attention to cucumbers (not the vines) that start growing
through the wire...and move them to on side or the other, once they start growing
through the wire it is hard to get them out or sometimes impossible without ruining the cucumber. A person can grow them up on almost anything that is sturdy. I once grew some in big buckets and made a wooden structure..like a saw horse over the buckets for them to vine on. So even when space is a problem, if you think about it you can come up with a solution.