We have the vertical version of those cellular shades for our sliding doors, which are set up the same way you describe (in an eating nook, leading out to our deck). We've been very happy with them, but I can't speak to the pet issue. I do know you can clean them using the brush attachment on your vacuum. Maybe pick a color that would camouflage the pet hair?
Hard to find a good pic online but here are a couple of examples. Ours are inside mounted like the second photo, which makes for a nice, clean look.
We have the same set up as you with a slider to the backyard off of the breakfast nook. I just have normal curtains and added the circle hooks so it can slide open and closed easier. I have 2 dogs. The curtains are tan with a design in them, so it hides any dirt from the dogs.