Greetings Excaliber.
Not the first, nor the last. El puelche shared in a prior post of a time back in the Pinochett era when the government was attempting to settle the Patagonian islands. Few took advantage of the offer, and since then, they have virtually all been taken off the market for sale, and in fact are declared property of Chile, and made into Park property. Few exceptions do exist, and these are in the Millions of US. Chile did an about face back in our Clinton years, and decided to grab land like our presidency did, and so now they have made most of the island chains parks.
With that being said, whether there still are some diehards out there from the original colonization efforts, I haven't any details other than what he briefly wrote, and that was an off hand.
Best that I can tell, and Admin can advise us otherwise, there exists the 50 year lease option like most park land in America. OF course how to get to where they will lease this land to you is another investigation, one which I would find fun in a running your head into a wall kind of way. See which one yields first. Of course, money may make the doors open, as to whether you can get ships to supply you...you're on your own. There are channels used in open navigation by ferries, so I would focus on building/leasing along one of these corridors. Satellite internet and phone will be your only options. These aren't cheap, though the price of phones is coming down to a few $1000, and they have worked out text messages for free with you $2 a minute rates. Not exactly T-3.
I haven't worked out the satellite internet bugs yet, but there are emerging services that aren't to astronomical

, but you are investing in technology that is unproven. Worse, hasn't even passed market acceptance, yet alone government approval. So early adopter will likely be your only option, besides text messages only with Satellite phones on the Irridium systems.
As to being on your own in the Patagonian Temperate Jungle Islands, or further south in the Mountainous Wilderness and Glacier Pack, you will be stepping off into uncharted territory. The sheer unknown, vividly wild, and uninhabited awaits you. You aren't completely off the planet, but you're getting a site closer. What prehistoric relics still walk, swim and inhabit these haunts, who knows? There are plants here that simply aren't in any known horticulturists repositories. Oh, and you will have plenty of bamboo to deal with. Islands literally covered with them. These will make agriculture difficult to start, and means you will have to clear cut an area to begin with. Those without bamboo, will likely be completely overgrown in trees, some of which the country deems invaluable and on the verge of extinction. Sacrifice the wild to make your sense of order.
Better rent a launch or skiff, and spend some alone time finding what would be an habitable and arable landscape, then having to rip up and tear down the jungle.
Some of these islands are beyond inhospitable. Most of these are fjords. Think Norway, with little in the way of level. Mountainous with some vegetation. You would be reduced to daily survival, and far from rescue for a long time before any beginning could be achieved. Bring earthmovers, tnt, and a sea plane when you need civilization.
I recommend trying Chiloe or a little farther to Melinka, or still farther to Tierra del Fuego. Of course, you'd be going from Seattle, to Vancouver to Alaska, and so, figure on what to bring with you based on where you search.
Just sit right back and you'll hear a tale, a tale of a fateful trip...G out
PS-Who hasn't dreamed of dropping out and holing up on his own private island? A dream, a possibility, and certainly not unattainable. Just not sure on this Patagonian locale. Although isolated, desolate, and mostly deserted, yet not a habitation of pirates like the South China Sea. Hope you'll keep me up on your searches.
PPSS- A German Battleship, the Dresden, hid in the Patagonian Coastline back in WWI. Wasn't there another German Battleship that did the same in WWII?