There are indeed miles and miles of miles and miles. Problem is, very much of it is in-accessible by highway vehicle, and trapping along most roads will already have trappers. And trapping along another trapper's line is severely frowned upon. There are trappers, myself included, (for decades, while I was there), that access more remote areas by boat. Most, not all, of those areas that trappers use boats in are in Southeast Alaska. There are only localized roads, as other than Haines and Skagway, all other towns and villages are locked away from any road access. To get to these places you either fly in or take a Ferry or large fishing boat, (think 42 foot minimum, to the crab boats you see on Deadliest Catch). It is about a 3–6-day ocean crossing, starting typically in Bellingham, Washington. Once you arrive to whichever post you choose, you will encounter trappers. They will not want you anywhere near their line. The local road systems will definitely have trappers that would consider the areas they have been trapping "their" line and area.
The interior is gigantic, and also has few roads, proportionately speaking. And once again, you will find other trappers. And accessing that vast country requires serious thought and planning. Alaska wilderness is no joke. Machines are really your only chance to find your own place to trap, and an airplane will get it done. It's also an entire subject all in itself. Suffice it to say, a day trapper does not use this machine or trap areas that require such travel. Snowmachines will get you into the outback, but if you are looking to have your own trap line, you will likely need to spend a summer cutting a trail first. Several members here can provide information on what that involves on the Wilderness Forum.
Then there is REALLY REMOTE: These areas I consider extremely remote would likely get a trapper from the lower 48 killed the very first month of the very first season. I'm talking about the outer coast of the mainland, west of Cook Inlet, with a few exceptions. And those exceptions would include help with logistics, cooperation from the various villages, and a fat wallet. And, yep, machines. And again, you will find trappers near any civilized areas. These areas are very expensive to get to, and getting all the gear and supplies needed to run a line for one season would certainly leave him many thousands of dollars in the hole.
The difficulty of the terrain in most of Alaska simply can't be over-stated. In Juneau, where I trapped for about 35 years, (also a few years on Kodiak Island), every year several people visiting on cruise ships perish while simply hiking. Almost always, they see an interesting feature and get off the beaten path and either tumble down a mountain or disappear. Some are never found.
Southeast is THICK!!! And STEEP!!! The water is extremely cold and dangerous, and during winter, much more so. Then there are the tides: In my old area there were tides in winter that went from -4 feet or more, to over 22 feet. That is 26 feet of tide move in one direction or the other in 6 hours. Most of that moves in the middle 3 hours. You can't hit the beach, toss an anchor and mosey into the woods and start walking your line. If you do and the boat, (or in my case skiff), goes dry, you are in serious doo-doo. It might be 12 hours before you float again. You might have 1/2 that much time for daylight. If you left the ramp in the dark. And the waves might make a beach landing impossible. And walking back to the nearest road in winter may well be impossible. And even if you somehow do make it to a road, it might be days before somebody actually drives that road, or it gets snowed in.
The interior brings equally dangerous challenges. All can kill you. And you get hurt, even getting your machine stuck can be life-threatening. Without a partner, AND a plan, things get serious fast.
I don't know enough about the Western Outer Coast areas to include every challenge, other than what I have learned from those that have been there and done that back when I lived in Kodiak, (early 80s), and current friends that do that now, (Otterman, on here, for one), and know some of the challenges they face even while living there.
Then there is fur. Contrary to popular belief, most areas are hungry. The country, and the animals that inhabit it. Guys in the interior that put up 100 marten a year might have a line 200 miles long, and multiple line cabins to pull it off. Those that trap Southeast by boat? Every single one will have near-death stories for you. Some aren't here to tell them, anymore.
This is a long explanation of why open public land for laying some steel in the Great State of Alaska for one and all isn't what those from down south think it is. And without actually living there, learning the lay of the land and the animals that use that land, is a fool's errand.
I don't mean to discourage the dream. Trapping Alaska and the incredible animals that go with that is a dang fine dream to have! I had that dream growing up in Oregon as a kid. I made it happen in 1982. It was the best thing I ever did, living that dream. But I couldn't have pulled it off without a little help along the way.
Below I will include a link several of us from the Wilderness Forum put together that we thought would be helpful to those with The Dream.