As for being "feasible", why wouldn't it be? From a strictly physical, "is it possible to construct such an item" standpoint, of course it is. We've built everything from hydroelectric dams so massive they're altering how the very tectonic plates themselves move, to transistors so tiny we could etch about half a million of them on a single red blood cell.
Presuming this gun isn't supposed to fire "Eraser" style 'just short of the speed of light' bullets or Star Trek phaser beams or something, the engineering is easy, straightforward, and basic.
And as for recoil, it's not as bad as you might think. Both of those homemade revolvers above were almost certainly quite shootable, simply because they're heavy. A typical 12-gauge shotgun might weigh 7-8 pounds and have a bit of a kick to it, the one revolver is listed at 11 pounds, which would damp it even more. Add a bit to bring it up to say 15 lb or so, and while it'd take a bit of strength to hold it up long enough to fire it, the recoil would be manageable even by a normal human.