I have found that they will last a good long time in the fridge as long as they have some air circulation. A few years ago, I cut some air holes in the lid of this container that we always use to hold grapes.
It's been working well until a couple of months ago, when a whole corner of the lid cracked off.
That made it just too trashy looking. I thought I'd buy a new container to punch some holes in, but then I got the idea to just sew up a fabric "cap" for this container.
I cut a rectangle, a few inches larger than the lid, then sewed a deep hem in all the sides, leaving an opening to run elastic through.
I put my elastic on a safety pin and fished it through the hem, then knotted the ends.
I was thinking that I should have rounded off the corners, but actually those square corners create a nice little tab to grab to pull the cover back.
It seems to be working fine and it will be easy to keep clean with a quick wash.