The question is about living on land that is not owned by a private individual in the USA, and is not designated as a state park or designated wildlife refuge, which have more restrictions.
It's not clear what sort of land you have in mind, and if such land even exists.
- On land owned by individuals, this is generally illegal. The basic offense is trespassing; if you attempt to build stuff or chop down trees there may be other charges depending on specific local laws. There are also sometimes squatters' rights laws that grant you a claim after living there for a long time, but none of them apply on day 1.
- On land owned by private organizations it would be the same story as that owned by individuals, you would just get sued by Acme Inc. instead of John Doe.
- Land owned by various government agencies is similarly protected. For example, if you sneak into a military base and camp there, it will not go well. Certain agencies, such as the Bureau of Land Management (BLM), Nation Park Service (NPS), Department of Agriculture (USDA) and others choose to allow the public to use land they own. This will usually come with certain restrictions which clearly spell out what you can and cannot do, and you are assumed to agree to them by entering the area. Almost none of them allow you to take up long term residence or construct dwellings without explicit permission.
As an example, most NPS parks will say you can camp up to 14 days at a time. How and where you can camp is restricted to some extent, but usually the restrictions are reasonable and you can camp in the backcountry in most parks. I believe they don't mind it if you camp at park A for 14 days, then camp at park B for 14 days, and repeat - of course you won't be able to indicate the park as your permanent address on tax forms and other documents. Typically parks will have a "camp host" who is allowed to live there long term, in a tent or RV, in exchange for some minor duties that help manage the campsite. This sounds closest to what you describe, but of course it's something you officially agree on with the park, you don't just show up and do it. NPS would not normally allow you to build a cabin (if you are an employee of the park they may let you live in an existing cabin as part of your contract), and if you have an RV it would have to stay in a designated RV area, not wherever in the park.
National Forests and BLM lands sometimes allow you to just go and camp without making a reservation or paying for a campsite. Like parks, they will restrict where RVs and other vehicles can go, and they will not allow you to build structures.
Funnily enough, some National Forests do allow you to harvest wood (chop trees). There is a limit to how much, what you can harvest, and you may be required to get a permit from them first. The permit is not necessarily a complex thing, often it's just a paper slip you get from the office for free or a nominal fee. I believe there are some that do not require any permit (but you do have to follow their rules).
You can also hunt and fish on many public lands, but typically you will need to have appropriate permits. Sometimes these may be bundled with the camping permit, but I've never heard of any government owned land where you can hunt without any sort of license at all.
Your question sounds more like permanently settling rather than living temporarily like hikers, campers, hunters, backpackers and so on. There is almost no land in the US that is not owned by some entity. Almost all land that is owned will not allow you to live there permanently (without a contract) nor build permanent structures. This is probably because allowing people to live permanently on your land and build structures there (without a contract) effectively transfers the claim to them, so nobody would do it any more than they would just gift you the land.
Suppose you left your cabin door open by mistake and someone else came in and doesn't want to leave your cabin. What would happen if you called police?
The police would come, take everyone's statements, determine that both of you are (I'm assuming) trespassing, arrest both of you, and both of you would get sued by the owner of the property. You may have some limited ability to sue the other guy for using the property you had in the cabin, but this is underminded by the fact that you didn't even lock the door. You don't get to evict him from it because you don't own it (unless you can demonstrate otherwise) - instead, both of you would get evicted by whoever holds the title to the land. They may or may not be required to return the personal belongings in your cabin. I doubt they would have to return the materials of the cabin or compensate you for it.