I want to go camping in the wilderness on my own, without anyone else around for miles. Is that even possible? I know a lot of national parks have designated areas, but surely they can't enforce that over 100s of miles of widerness? I have done this in the UK, where I could camp deep in the forest for a week with no-one else around.
The short answer is Yes, but.
I will confine my answer to the California Sierra, because that is the area I know best.
You can certainly camp for days without seeing anybody, except possibly in the distance, even in Yosemite. And it is easier to get far away from people in the Eastern Sierra, starting, for example, from trailheads outside of Bishop, CA.
You will have to leave the trail and go cross country to find this isolation, and a wilderness permit is required for overnight hikes. If you are off trail, the wilderness permit does not place restrictions on where you can go, or where you must camp, except for restrictions on how far your camp should be from lakes and streams, the elevation at which wood fires are verboten, and perhaps a few other requirements, but they are mostly environmental requirements, and we have never found them onerous or interfering with the desire for isolation.
For example, Humphreys Basin will give you great solitude, plus a multitude of lakes, large, medium and small between 10,000 feet(3,048 meters) and over 12,000 feet in elevation. streams everywhere, a 14,000 foot mountain, and few trails. Few trails mean few people. You may see a pack train going over Paiute Pass to Golden Trout Lake, but you can steer clear of that.
You may not be literally miles from the nearest person, but the terrain is such that being even one mile from the nearest person gives you the same feeling.
Wyoming and Canada should have even more isolation, but there you may have grizzlies to worry about, whereas the Sierra has only black bears.
Yes this is possible. I think it is fairly easy in the western US especially during the winter if you're willing to work hard. An obvious, albeit extreme example of this is the state of Alaska. Alaska has a population of roughly 740k people and an area of roughly 670k square miles. Clearly in a state this big with a population this sparse a dedicated person could disappear into the wilderness for a long time without seeing anyone. One book you could read about someone doing this with grave consequences is Into the Wild by Jon Krakauer.
In summary to accomplish your goal go to large wilderness areas that aren't frequently visited during times of the year when it is the least popular to visit. Alternatively go somewhere like Alaska that's too big and brutal to be overrun by people.
In the east there is less wild land and more people exploring the natural areas that are available. I'm big on pathless forests, and to find these places I've learned to either seek out privately owned forest properties with welcoming landowners who don't mind my kind of wandering and bushcraft activities, or to go to state forest which is the crux of my answer.
State forests in the east are not as intensively managed, advertised, or visited as state parks. State forests are not quite wilderness in that they are usually managed based on human-defined priorities, be it sustainable timber production, wildlife, recreation, unique areas, or some combination. The wildlife and timber production areas are generally the least obviously tread by humans and the most friendly to your own activities (i.e. you can camp there, following certain rules and conditions). There are also areas designated as different degrees of wilderness such as in the Adirondack Park. You can use those designations and maps to seek out the more wild areas. It could also be worth looking up recreational stats for different areas, as some wilderness areas (e.g. Adirondack High Peaks) are far more visited than some state forest recreation areas.
If in doubt, ask the local environmental agency about location recommendations and what the rules are with camping. In NY, most state forest can be camped on as long as you follow some rules and aren't in specially designated unique areas. In my experience state environmental employees love the outdoors and are happy to help you enjoy nature, so I encourage asking for local tips.
The parks are quite regulated, and for many you need a permit for back country (random) camping.
The wilderness areas are more liberal. They like to know where you intend to go. Note however that in a lot of them the trails have not been maintained for decades.
The national forests are not as spectacular for scenery, but many of them have extensive trails that aren't used a lot.
In Canada I no longer use the National Parks. The red tape is too extensive, and the fees are expensive. The trails are well maintained, and too often you will meet someone coming the other way every 15 minutes.
Instead I use Willmore Wilderness Park. Once you get a day in from the trailhead seeing anyone is rare. Last summer there were two sections where I found bootprints, and 3 sections where I found horse droppings. Given the weather either could have been up to a month old. I have never seen a ranger or a conservation officer in Willmore.
Willmore has both bears and wolves. Both are unaccustomed to people and keep their distance.
In the US, Designated Wilderness Areas are areas relatively untouched by human influence where users are only allowed on foot and with pack animals. Some areas will have other restrictions, like permit requirements for overnight trips or seasonal fire restrictions. Most are tens of thousands of acres, which is plenty big enough to get out of hearing range of other humans for as long as your supplies hold out.
To camp in these for weeks, look up the local land management office in charge of them and give them a call to learn about access, current conditions and restrictions. They'll know what access points are accessible, what ones are washed out, and which areas you can visit to have the least chance of being seen. The relevant agency can be the Bureau of Land Management (BLM), US Forest Service, or the National Park Service. You'll want the local office, not one in DC. There's a list at https://en.wikipedia.org/wiki/List_of_U.S._Wilderness_Areas.
In addition, in the western US, there are huge tracts of unprotected, undeveloped government owned land managed by the BLM that would fit your needs perfectly, but the best places are harder to find through internet searches. If you get to know local outdoorsmen and women, they can give you better direction for specific places that would suit your needs.