OK, now I understand better. Assuming that you can get all the weight off the 3/16" cables while taking them down or putting them up, and assuming that the turnbuckles are long enough to take up 0.5" plus however much you need to clear the end of the hooks, which I guess would be another 0.5" to 1", then you'll only need to provide a 20-30 pound sidewise pull to release the hooks. Should be pretty doable.
Also, if the nets weigh 90 lbs, or 30 lbs per cable, I calculate that you need to provide 9" of sag to keep the sideways force on the screw eyes to about 300 lbs when the nets are suspended. So be careful not to try to get less sag.
The other thing to worry about is if the building is all concrete, and therfore very rigid. As the temperature changes, the cable and building are likely to change length at different rates. That may stretch the cable or fittings, and then if the temperature returns, you'll get a larger sag. In a wood framed building or one with steel studs, this won't be a problem because the wall will just bend. For example, to reduce the tension from 1000 lbs to 300 lbs, the walls only need to deflect by 0.004 inches!
Adding stable supports in the middle of the span will significantly reduce the tension for a given sag, but it doesn't look like you need that. If the top of the supports can move in a direction perpendicular to the cable, I don't think the tension will be reduce much at all. So it's probably more trouble than its worth.