By comparing markings, serial numbers etc to known records from gun makers.
Old enough to have money and see over the counter.
Gun shop, gun show, on line auctions
The 8 or 10 barrel Gatling gun was the early machine gun. See: http://en.wikipedia.org/wiki/Gatling_Gun
Old gun catalogs
Old gun catalogs and magazines
Be old enough, not be a convicted criminal, druggy, illegal alien, have two forms of ID and the money to pay for a gun. Go see a gun dealer.
see the link below for UK air gun laws
On line auctions are the best place to see what they are bringing.
Earlier gun than the Old West.
date a gun
You need to contact your local police to see if there are any local laws regarding this.
It looks like an old Colt 45. See the link below.