Pull all U-nails holding the barbed wire strands to any of the wooden fence posts along the length of the fence. Use a pair of fencing pliers to eject the nails from the wood. Do this by gripping the nail with the grips of the pliers and prying the nail out of the post.
Unravel all wire clips holding any of the strands of barbed wire to the steel T-posts along the length of the fence. Use the fencing pliers to grip the ends of the clips and untwist them from around the wire. If you can't untwist them, cut them off instead.
Go to the corner of the fence line and cut the top strand of barbed wire with a pair of wire cutters. Grab the end of this strand and loop it into a 3-foot diameter hoop. Roll the hoop along the length of the fence while wrapping the stand of barbed wire around it. Weave the wire back and forth across the circumference of the hoop to keep it together. The barbs do most of the work at holding the wire together. Snip the strand at the other end of the fence line and wrap it perpendicularly around the hoop of barbed wire fencing.
Cut the second strand beneath where the first strand was with the wire cutters. Repeat the aforementioned looping, hooping and rolling process until you get to the other end. Remove all strands in this manner. At this point, only the fence posts remain.
Wiggle the steel T-posts back and forth until they loosen and pull them out of the ground by hand. If you can't get them out, dig around the base with a shovel. Do the same with the wood posts. You may have to dig first and wiggle second to remove the wood fence posts.