Is it possible to design a GA plane with a 5' -10' wing span?
They can, its just not a very efficient size for one that needs to cary people. The simple answer is that everything comes with a trade off, there are plenty of planes with small wing spans like this one, and this one and some planes that don't have wings at all... But smaller wing spans means a thicker wing, at some point you will run into a drag issue, a roll stability issue or both. 

When it comes to aircraft people tend to be concerned with efficiency as air travel has always been about going far, fast, and for a similar cost to what you can on another means of transport. As such efficient designs win out over novel ones or ones that provide some means of limited added benefit. 

On any note most of the supporting infrastructure has been designed for planes with substantial wing spans so aircraft width in the GA market is not really a huge issue. Saying a narrower wing is "way more useful" depends on your use case.