Can you draw a floor and with measurements and the dimensions of your furniture. My idea would be to move the furniture around.

My starting point would be to keep the table parallel to the iron table but move it towards the left wall. Then use the tall cabinet and place it in front of the table with doors facing our way so the cabinet acts like a wall. To do something like that, you would need to move the rest of the furniture around but without seeing a floor plan, I can't suggest where to move them.

Velcro and a gathered skirt of fabric around table edge.
Have an opening in skirt where access is needed.
http://houzz.com/photos/75494

I would get rid of the table and get some stock white kitchen cabinets and surround with MDF painted white with a melamine counter top. This would give you more storage and get things out of sight.

your table is great. do you use it for folding? I would keep it near the dryer. I don't think you need to hide your sorted clothes, it is a laundry room :) but the curtain is a nice idea. do you keep your cleaning products in the cabinet? things should be handy for you and easy to lift (or not) Don't worry so much about what it looks like but how it flows for you as you work :)