Marry into the warden service, kiss the governors backside, pay-off a legislator. . . . you get the idea. Some folks get lucky and have had several moose permits.
I have put in every year since 1980 (start of the moose hunting in Maine) and have not been picked yet. My wife put in every year and finally got one in 2002.
If you really want to shoot moose, Quebec lets you shoot two for around $600. Just buy the ticket and hire or take a guide with you.