Forgive me for suggesting the OBVIOUS, but why not have someone who is at least slightly artistic create the web by tying plain old string or monofilament fishing line into the web arrangement?  The string would be more visible and probably more manageable.  To make it quick to put up and take down, put a simple loop or hook on the four "corners" to attach to your trees.  You could roll it up on a paper tube (after folding it over once or twice) to keep it from tangling between uses.  (Handle it much the same as you would a volleyball net.)  Just a suggestion.  Good luck with whatever you end up doing!

plain string will work better than fishing line. Get your dimensions and stake them out in the grass with rebar or tent stakes that stick up 6 inches above the yard. Tie string across like a pizza pattern to opposite rebar. tie the center together with one string. Now work from the center and make your web by tying a knot on each string in a circular pattern. It is best to use a knot like a clove hitch. Those can be adjusted on the string incase you pull one too tight and it sags another line. After you get the web how you like it, take a hot glue gun and glue each knot into place so they will not slide or adjust as you take it down and put it back up. Now put hooks on the trees or nails even and  tie each end to the nail. You may have to tie a runner string on the top and bottom between the trees to tie the outer edges of the web too.