Is it possible to design a GA plane with a 5' -10' wing span? We see sparrows with 6" wing span land on a dime, yet it seems there no GA planes you can buy or build from a kit with short bodies, short wing spans, 
something with a 20' landing/takeoff roll, something with folding wings you can drive out of your garage, take off from your driveway ( if you live in a non built up area) and land almost anywhere at 20mph. 

It seems like GA is dying, but I think a lot more people would be interested in flying if GA planes were way more useful.

Yes, there was the BD5 which was nice and small, not STOL though, the Alaskan bush pilots are taking off/landing in 10' ( yeah!!) in their super cubs, a standard Zenith 701 STOL which can take off in 25' with a good headwind, but all have large spans 

Using the Cri Cri as an example


Cri Cri Monoplane: , I think it has a span of 15’ and a chord of 2’ for a wing area of 30 ft2. 
Cri Cri biplane: Now imagine  converting it into a biplane with a span of 1/2 or 7.5’, same chord.
Cri chi triplane: Now take the Cri Cri monoplane and reduce the span to 1/3, or 5', same chord.


Why can't we have planes with 5-10' wing spans.

Is it impossible to design roll stability into a plane with such a short wing span?

Is there any other reason, like 50% effectiveness, etc. at low span due to span wise flow, instead of chord wise flow, etc.

Looking at Selig's work on multi element racing car wings, their spans are only 5', with a cl of up to 4.5!!  He does mention they need huge end plates to avoid span wise flow.
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.