I don’t know the name of them, but Lee Valley had some kind of hinge that allowed the door to do this -I think it was for media cabinets

They’re called flipper doors and they’ll take up several inches of width. Plus, not ideal to enclose the machines.

The example you show is kind of absurd: all that trouble to conceal the units, but the tile backsplash spells out “Laundry?” Why are people so embarrassed about the very common task of laundry?

Better to have a closet for both of them and any/all laundry related items than to try to "hide" them behind cabinets.

EZ Pocket Door System-Pocket Door Slide

https://www.rockler.com/ez-pocket-door-system-pocket-door-slide

Thank you everyone! If you all saw the room, you would understand why it has to be done this way from the homeowners point of view. To each their own.

I agree with the louvered door suggestion to allow moisture/steam to escape.

cabinetparts.com also has the type of hinges you'll need.

If the homeowner doesn't want louvered doors, you could use solid doors that meet the ventilation requirements specified in the appliance installation manuals. Of course, check the those manuals for required clearances. I'd make the top removable to make it easier to access the plumbing and insert/remove/service the appliances in the cabinet.

-- amateur