I have heard that onions and potatoes shouldn't be stored together. Why will this make the potatoes spoil faster?
As you can see at the Chapter 7: Storage of horticultural crops of the book PDF: Small-Scale Postharvest Handling Practices: A Manual for Horticultural Crops (4th Edition):
So, if you store them together, either one or the other will be in bad storage conditions.
Anyway, storage of horticultural crops cannot just be resumed by "don't put this or that together", as you may have seen:
If the subject interests you, all this (and many other things) is pretty well explained in the following references:
Ethylene is also a "critical factor", and this topic is more developped in the following resources:
When stored together, onions release gases that will cause the potatoes to spoil faster and become mushy, if you will.