"Hidden" door with drywall

One idea would be a curtain.. Maybe you already have some on the windows.. then I would use the same fabric.. Or something very light.. like the staircase. With curtain ties, you can create very decorative folds so that it doesn’t look sloppy.

You can choose a simple 'tapetentüre'; for example, it can be hinged with piano hinges (cheap version), with invisible door fittings, or with furniture fittings (e.g. kitchen furniture). Then you either choose push to open or a discreet pull fitting.

Or a shelf sliding on rails... (if there is space next to it)

I can also think of a nice blind; that way we have just hidden the washing machine.

Thank you for all the ideas. We hadn't considered some of them at all and can now see how to solve it. The goal was actually to make the wall as unobtrusive as possible.