Building your own airplane is possible and actually quite common. There are kits for anything from one-seaters with open cockpit to pressurized turboprops.

http://eaa.org/homebuilders/

No I meant like newer generation planes with more efficient engines, lighter, more aerodynamic, etc.

I'm not sure if current airplane designs are as good as they can...

True, engine efficiency is kind of flat-lined. Turbines aren't efficient for most GA operations, but there are new diesel aviation engines coming online.

Aerodynamically, I don't know about big gains. Something like a Lancair Legacy is pretty low drag. You can make lower-drag planes, but that typically comes with much higher stall/takeoff/landing speeds that make the plane harder to fly and much less safe in the event of a forced landing.

Personally I'm looking forward to the next generation of electric aircraft, once battery density improves and costs come down. A Chinese company is already producing an electric ultralight and working on a small GA electric: http://www.flyingmag.com/aircraft/lsasport/germany-certifies...