We use a hanging screen with magnets in the middle. You just walk right through it and it reconnects. It actually works really nice.
It's like this.
http://www.bugoffscreen.com/But I'm sure you could make one.
It's not even permanent, it's hanging from a spring rod so I can take it down if I want.
I figured I would try it before making a door and so far it's been there 3 years now.
Whether that's because I'm lazy or not is undecided. I did make the side walls which is the hard part I guess so a door at this point would be pretty easy.
I would have made the screen door with a piano hinge down the middle and made it a bi-fold. I may still, we'll see.