"One" must have a lot more than a"boat" and a whim. First you must be actively involved in commercial fishing as your full time occupation for a few years and be able to prove it. Then you might get a National Marine Fisheries License , then you can apply to the Alaska Commercial Fisheries Entry Commission.
Alaska's Commercial Fisheries Entry Commission has nothing to do with the issuing of quota. The CFEC only issues the little plastic cards that allow the product to be delivered in Alaska and it allows them to fish inside of three miles, which is very unlikely.
One can walk right into any NMFS office and obtain a federal waters fishing permit at any time. However you will be limited in what you can catch. Many of the more lucrative species are either under moratoreum (no new entrants allowed) LLP (Limited Licence Program) or quota. Having proof of fishing for a few years might get a good laugh, but will not get you any quota.
To obtain quota you need to find someone willing to sell theirs.Anybody with a LOT of money may purchase quota.I think the price for King Crab has been quoted to be around 30 dollars a pound. HOWEVER, currently there are no sellers. There is some amount of temporary trading going on among the quota share holders, but there is no Bering Sea crab quota on the market.