While the pulsejet powered [Kawanishi Baika][1] from the WW2 was really far from the GA, from the picture it looks like it was a relatively small plane of very comparable size, capable of taking up under own power, manned. While none were eventually built, seems that the idea was accepted rather seriously at the time.  (Its sister ship is the Fieseler Fi 103R, of which 175 were built and a few were test-flown.  But they launched by dropping from a bomber, which avoided the ground-launch disadvantages noted in Peter Kämpf's answer.  Again, not practical for light GA.)


  [1]: https://en.wikipedia.org/wiki/Kawanishi_Baika