Has anyone found a good strategy for storing their cable extensions that doesn’t take forever to do and doesn’t make a bird nest?

Give us a little more basic info. Are you meaning extensions for snares separately, or extensions for earth anchors or for Dp's? Have a little different way for mine (like a lot of fellas on here do with theirs). .......................the mike

coil em like snares and 5 gal buckets work for me

Usually 3 to 5 foot long cable with loop on each end. I use them for DPs and for body grip traps.
I was thinking about wrapping them on a piece of wood with two notches on each end to put the loops on. Maybe that’s over thinking it though, and I should just coil them like a snare like sniper said.

Coil and tuck the ends in, like you would a snare. A bucket will work good, but if you want them in smaller packets, put a dozen or so coils in a gallon ziploc bag. I used to use them extensively for marten bodygrips, and still do some, although my newer boxes all have a short chunk of cable with a loop on the top of the box and I just use a carabiner to snap the trap chain to it. But I still carry some extensions for older boxes and stuff. Just grab a ziploc or two and toss under the snowmobile seat. The bag keeps them clean (not an issue with marten anyways) and prevents the ends from coming untucked.

I like the smaller 2 gallon buckets.

Keep snares and extensions coiled and in plastic tool box....

My cable extensions are used with beaver snares. They’re 7-10 ft long. They get coiled like I do my snares. I can store almost 40 in a 2-gallon bucket. Don’t over-think this.