OK, so I have high tensile and the top wire is coated, and they’re all staped into wooden posts. We have done several projects that involved removing one or more wires from a section of the property. All told I’ve taken down probably 5-6000 LF of wire.
First take the tension off the fence by backing off the tensioning ratchets several clicks.
It’s not at all difficult to pop the staples with a good pair of fencing pliers.
As others have mentioned, dealing with the wire is going to be the biggest hassle. We have not bothered trying to wind it back onto a spool. For sections where you’re getting rid of the old wire, it’s easier to just wind it into loops on the ground like a garden hose – stand on the coil on the ground with your foot as you make the next loop. Every 50-100ft or so, cut the wire and secure your loops with a zip tie or two. Leave the coils on the ground then drive the fenceline with the ATV trailer to pick them all up. If I’m going to be re-using that wire, we’ve done this same “coil it into loops” method for much longer lengths. You can still apply a zip tie for every 100ft or so of wire to manage the coil, then keep coiling. And if it’s too unwieldy, just cut the wire and deal with the fact that you’ll have to do some splicing when you reinstall. Splices are not that big a deal. (though stripping the coating in order to get to bare wire is a PITA.
IMPORTANT SAFETY RULE: wear good gloves and safety glasses or even a face shield. The wire can and will whip around towards your face, much faster than you can react. Only thing predictable about stretched, old wire is that it will behave unpredictably. I can’t stress this enough.
I’ve actually done most of this work myself-- it’s not particularly grueling work. More tedious than anything. And when Mr. Essex has helped, no divorce was ever threatened 
You can totally do this.