Shotguns with barrles under 18" are in the same legal category as machineguns, as defined by the National Fireams Act of 1934 (NFA). To legally obtain one, you must fill out some BATF paperwork, get fingerprinted, get photographed, get permission of your local Chief Law Enforcement Officer (sheriff, judge, AG, DA, ...), wait a couple months for the BATF to process all this, and pay a $200 tax. For all this red tape, time, money, and possible hassles (CLEOs often refuse to sign - no signature, no gun), you get federal permission to possess the gun, your name on permanent federal records as "owner of really nasty gun", and must ask the BATF for permission to take it out of state.
Go hang out in TFL's "Full Auto" forum for a while; the focus there is on machineguns but is a catch-all for any NFA-related guns and issues.
Be aware too that many states prohibit short-barreled shotguns.
