Again aside from every sort of problem involving freedom, there would be a long list of things which the control freaks would start working at banning the day after they banned private ownership of firearms.
Algor and a large group of his fellow de-mokkker-rats would be right in there trying to get privateownership of motor vehicles banned, and you can be sure that they'd be out after privately owned archery equipment.
Knives would be banned, along with axes and hatchets. I mean, clearly, Jack the Ripper and Elizabeth Bordon never owned a gun between them.
Sex other than for producing children would likely be banned and the war on drugs would likely be extended to include alcohol.
"Archery equipment??" you ask....
Basically, anybody who's halfway good with a modern compound bow set at 70 lbs or over can hit a soccer-ball sized target at 100 yards, and it doesn't make any noise. The arrow will appear to rise six or eight feet over the height of the target, and fall into it.
In fact, if you ever took a modern compound bow set at 80 lbs and aimed it up at a 40 degree angle like you see the British doing in movies, the arrow would never come down. Or it it did, it wouldn't come down in your county.
Consider that AfterShock is now claiming to be achieving ten second kills on cape buffalo with their 125 grain mechanical broadheads; a 375 safari rifle can't do that. They've damned near made firearms obsolete for hunting. In fact, their web site shows a girl with a 45-lb bow and a 450-lb boar hog she killed with it:
http://www.aftershockarchery.com/