I keep my smaller nets sitting in a bucket of weak betadine solution, which I change about weekly. This was SOP at a fish store I worked at when I was a kid, although I am pretty sure it was bleach back then. Until recently I took the net out of the bucket, flicked off the excess moisture and then went to work with it. Having recently installed a small laundry tub in my basement, I now quickly rinse the net off to remove all betadine residue, but I know that's not necessary.
I've always got a couple of larger landing nets as well; they usually are carefully dried out between uses, easy to do since I don't have a lot of big fish. Back when I did, I would use the betadine solution but in one of those long, wide and shallow plastic storage bins intended to slide under a bed. A couple inches of solution would cover the whole net, but would only use up a few gallons of solution.
All my outdoor ponds and tanks have their own dedicated nets, so they don't get disinfected, just kept separately by each pond.
A warning: one of my friends bought, on my recommendation, one of those terrific rubber-coated landing nets that are very easy on fish scales and slime. He forgetfully left the almost-new net soaking in bleach solution overnight...and the next day it was a brittle, deteriorating mass that fell apart practically under its own weight. Don't know if that's still a concern (this happened years ago) but I have carefully avoided exposing that type of net to chemicals since then.
