How do I update those old brown slidings doors to the closets?

Other than paint/stencils, there's not many other options. Any added trim would interfere with door operation, as would adding handles which project.

I'd be hesitant to apply anything sticky to door's surface, as removal would be impossible, without damage.

Paint, new handles, and stenciling it. Or, you could replace them. I love the doors that open out :)

https://www.hometalk.com/diy/repair/windows/update-your-doors-from-slab-to-fab-9896912

https://www.hometalk.com/diy/windows-doors/doors/the-easiest-way-to-update-a-flat-panel-door-40241166

https://www.hometalk.com/diy/windows-doors/doors/upgraded-updated-interior-doors-31588793

We flip. I had some damaged doors so painting was a necessity after fixing. SO boring looking still. Then I had the idea of stenciling a Mandela pattern in the area a door knocker would be. Stencils are great for sliding doors because they keep the depth of the door the same. It looked really good, people loved how unique the doors were.

Take them down and put up double doors that open outwards. But it would be simplier to paint them. Just remember to paint those gold round door handles. That also really dates the doors.

https://www.apartmenttherapy.com/diy-closet-door-makeovers-36873423

Maybe add Mirror Foil and a trim.

Paint and add trim with some bold hardware. I’ve also seen washi tape on them.

You could wallpaper them. You could also paint them and then mask off specific areas and paint them a slightly lighter or deeper tone of the same color to make them look like you've trimmed them out with molding to add some interest.