Ok so half cut onions might smell up your fridge, but otherwise they're just most likely to dehydrate without being covered in the fridge for a week. Also onions can prematurely ripen other produce when stored in the same place. Whole onions store well outside the fridge for weeks, even a month. Potatoes are the same.

Sealed food storage is best though, so if you find some leftover plastic containers they sell takeaway food in, washing and reusing these can be a free option. Same with glass jars. Check goodwill for Tupperware too.

Lemons I leave cut in the fridge for well over a week and they just dehydrate a little on the outside, I've only ever seen one get a little mold, most things can't live in that acidic an environment.

Most other things freeze really well, if that is an option for you. If you do buy fresh, you can increase their fridge lifespans by washing and then thoroughly drying them. The best veg are ones you can buy with the roots still attached, those can be planted or kept in the fridge a long while with the roots kept wet.