We have considered that but were told it would have to be built out about 4-5" which would impede flow of foot traffic through the room. Does anyone have experience with using piano hinges on bi-fold doors so they can be opened up without being confined to tracks?

Are these new appliances? I hate to throw another monkey wrench out there but with a front loader, you may want to be able to keep the door open when not in use to prevent mold buildup and odors. I love my front loader but do find it necessary to keep the door open - fortunately my laundry room allows me to do that behind closed doors. Maybe if you are diligent about wiping out interior you can avoid that, but just wanted to mention it. I used to have house with washer/dryer in kitchen but it was a top loader, and I think it's a better fit for when you can't close off washer and keep door open. I don't know how to solve your door problem but I would suggest installing a counter over the washer/dryer - we just have a low budget formica one in our laundry room and it really increases usable space. I see you are heading that direction with what looks like a piece of wood, but installing a counter will prevent things from falling off the back/sides. Good luck!

I would install a barn door, see Sliding Door website, or embrace the area and leave it open. By embracing you would need to have finish grade shelving and put attractive things on the shelves making sure you error on the more lean side.