I was wondering if anyone was aware of a way that magento would allow you to list your products similar to the following website:
(Basically, where the customer can type in how many of each size of product they require)
Yes, possible let's use group product. Group product already provides functionality to add multiple simple product into the cart.
For display propose you can create one attribute like color and display from it.
If you want to do using the configurable product then little bit hard to implement it.
Hope it will help you.
If it will help you then mark as solution.