After trying too many different options, I decided that GreenSavers [Amazon link] best met the twin goals of keeping veggies fresh while making the fridge navigable.
We have a relatively small model and it's typically overflowing with leftovers and wilting produce: delicious things would be lost until it was too late; sinister matter would be found instead. So we tried organizing our fridge with boxes, starting with basic peel-lid Tupperware and working our way up. Here's what we found.
• Basic boxes are excellent for organizing, but bad for visibility and only good for certain types of food (meat, cheese, leftovers) that need an airtight seal. Some brands (like Rubbermaid) tend to make theirs in annoying shapes. Dollar store models are cut from material that's either too brittle or too squidgy for the job, and, well, who knows what they're made of.
• Cheap "sets" are bad. One Amazon bestseller was so cheap-seeming I think it's just a desk organizer set being sold as a produce organizer, and won't even link to it ironically. Some don't even have lids. Don't be fooled.
• Attractive rustic glass sets with big corks and such: not for fridges. Great for fancy teas you never drink.
• The popular "Good Grips POP" organizers are expensive, of questionable mechanical durability, and are generally too tall for the fridge. They are freezer-safe, though. The bulky, clicky mechanical lids are just too annoying and unwieldy for them to work well for produce-keeping purposes.
The ones we settled on were the GreenSaver Produce Keeper range, also from Oxo.
They're similar in design to the Pop models, but have simple sliding lids to either close them or leave slats open. They stack. They're square. There's a model designed to slot nearly to the back of a normal-depth fridge. They're transparent. What distinguishes them are (a) an interior basket that keeps the produce from going nasty against the edges of the box, and (b) a filter that supposedly reduces the presence of ethylene gas that accelerates rot.
I can't speak to the filtering claim, but the basket works and in an obvious way: an obvious reduction in floppiness in the short run and even when you do leave stuff in too long, it doesn't get coated in plant slime.
There's a little illustration inside showing what produce keeps best with a little air.
Two caveats: they don't make a good airtight seal (though that's what Tupperware's for) and they say you should replace the filters every 90 days. At $10 for four official ones, that strikes me as a big nope. You can buy a 2.5lb tub of activated charcoal for $12 that will last forever.
Finally, we haven't tried the produce set from Princess House, another known-good brand. They don't bother with the filters, but have more elaborate (and less efficient) internal colander basket setups.