It is a little hard to tell from the photos, but it appears that a sliding door would likely not work.

For a sliding door to be most effective, you need the same amount of space on one side (or half on each side) as the opening. So if your opening is 4ft wide, you need 4 ft on one side OR 2 ft on both sides.

If you are close, say you have 3'10" on the side instead of 4, you might be able to do it, but the door would overlap the opening by a few inches when opened.

You could do a cool pattern curtain like shown above. But for better sound absorption I would try to find a better material. I am not sure if in your budget or plan but installing a pocket door would work well for this space.

Yes I have thought of a pocket door, I just did not know if I had the necessary room for one

I could not live with that. I do not like laundry facilities in the kitchen anyway & if you cannot open both doors fully, it is almost useless. I agree if there is any way to have one pocket door or two installed w/o costing you an arm & a leg, it's curtains for you (no pun intended).

I can open the doors all the way, just not the washer / dryer door all the way - I do not like the laundry there either, was only thing I did not like about house. I do not want a curtain, will look into pocket or sliding doors of some kind. The former owners had top loading and smaller washer/dryer.