Niihau is a privately owned island in Hawaii where the inhabitants speak Hawaiian. You can’t visit without an invitation though.
Nikolaevsk, Alaska is a community of Russian Old Believers. It happens to be not far from Homer, as mentioned above.
For a true end-of-the-road experience I guess you could drive to Deadhorse, Alaska on the Dalton Highway. There’s nothing very scenic or inviting once you get there though.
Arock, Oregon, like most places in SE Oregon, is fairly remote. But it was founded by Basque sheepherders. Still quite a few around. So, a bit remote and with a touch of the exotic (if you consider European language-isolate speakers exotic).
Not too far away is the gravesite of Jean Baptiste Charbonneau, son of Sacajawea, born during the Lewis and Clark expedition.
The area is “Owyhee country”. Some of the earliest miners came from Owhyee, aka Hawaii. But the Hawaiian culture died out. Too bad. I bet Basque-Hawaiian cuisine would have been interesting.
Lajitas’ most famous resident: Clay Henry, the beer-drinking goat (now deceased and stuffed).
Cedar Key is very near the Florida Keys; it’s one of them. It isn’t near the southern Keys, true, but it is an island with a limestone base which is the result of a coral reef and changing sea levels, which is what defines a key.
It’s also a beautiful little island with excellent kayaking/canoeing and snorkeling/scuba diving opportunities.
Great Basin National Park is a strange treasure, although not a town. It is surrounded by desert but is alpine, it’s a mountain, with, like, snow. Also has some interesting caves.
I did that drive and you’re right. Not only is there not much there, you’re not allowed to roam freely. You’d be better off driving to Inuvik, NWT on the Dempster Highway. Scenic rivers, free ferry crossings, First Nation town at the top of the world.
There’s the Deadhorse General Store that’s pretty cool, but that’s seen in ten minutes. Then just the novelty of laying on a rocky beach counting ice floes. If you’re lucky on the way up you’ll see some Musk Ox or a Grizz.
Yellow Pine, Idaho. not the most remote town in the state, but it is pretty remote and smack dab in the middle of the forest and mountains. It’s beautiful and it has a harmonica festival every year in August. If you want to be there with thousands of campers and see some harmonica players go then. If you want to visit with the 100 or so full time residents, go in July. Do not try to go in January. You won’t make it in without a snow mobile.
Angle Inlet, Minnesota too, although the landmass is connected.
Stop in at Anderson Valley Brewing Co. for a pint! If the mood strikes, they have a frisbee golf course on the property.
I’ve wanted to go there for years…is the road up there as scary as I’ve read?
Get Boont Amber Ale too, don’t be an eesole. You can learn from the label, or just drink much and mumble and pretend you know Boontling. Bahl hornin.
It’s 400+ miles from Cedar Key to the Florida Keys. You may call that “very near”. I do not.
And other than you, I have never heard anyone claim Cedar Key to be one of the islands commonly understood to be the Florida Keys. Mostly because it’s just not so. 
I think it would qualify as a Florida key (lower case), but not a member of the Florida Keys (upper case), which was how he spelled it.
[Hee Haw/] SAL-YOOT! [/Hee Haw]
This technically fails the “in the US” part, but there’s the islands of St. Pierre and Miquelon that are the last remnants of the French Empire in North America. So you can get to France via a short ferry ride from Newfoundland.
If you want remote and abandoned, here’s a list of ghost towns in the US.
ETA: “Ghost town” as in the town is deserted, not as in the place is haunted
Not so remote, but a beautiful place: Camp Sherman, OR which sits near the headwaters of the Metolius River, one of the prettiest fly-fishing streams you’d ever want to see. Only a few hundred full-time residents living blissfully at about 3,000 ft., but gobs of tourists visit the place. There is a general store for basics, and a very good restaurant that is closed in winter. For supplies more than what the general store can offer, there is the town of Sisters and the city of Bend not all that far away.
In a similar sense, you can get to the UK via a short ferry ride to the British Virgin Islands from St. Thomas (US Virgin Islands), or two short ferry rides from Puerto Rico with a stop in St. Thomas.
Back in about 1976 I lived in St. Thomas for about five months. Every sort of dropout, misanthrope, and escapee from the rat race was found there. I was friends with a former Playboy Bunny, a stranded ship captain, a frustrated rock singer, several drug/booze addled types, a construction dynamiter with a serious alcohol problem, a daughter of a very wealthy island family who had given them all the finger: the list goes on. I liked it a lot, but I suspect it’s changed a lot since then. If you’ve got some money to live on and want to get lost, the VI may be for you.
Off-topic, but given that Quebec is also quite close, I wonder why they’ve never tried to join Canada. Or have they?