To further answer your original question;
No, you may not live for free on the same area of public lands for any more than 14 days. In fact, you can not "reside" on public lands for even a single day. You must reside elsewhere and only be "camping" on public lands (usually with a 14 day max). Can you get away with longer? Sure, but you risk citation or even arrest (extremely unprobable of course). There are areas that allow "long term camping" in public lands (Quartzite comes to mind) but still costs a few bucks for the permit to do so.
Basically, you are required to move to completely different public lands every two weeks, so 26 moves per year. This is till "living for free" minus that you will have to spend some money on gas. But then the reality is that you will not be self sufficient on public lands and will need to come into town to resupply anyway, so you will need to move for that anyway. So, resupply to handle 2 weeks out, and move to a new public land each time you need to resupply. 26 moves per year will allow you to chase the good weather and keep fuel costs pretty minimal.
The obvious (but misguided) choice is to live on public lands close enough to a town to walk/bike in, but I can assure you if you are that close you will be under much higher scrutiny from the Rangers (and other campers who want to share the land) and you simply will not get away with longer stays.
Here is an example of doing it within the rules:
Winter in lower AZ - tons of different BLM lands to stay on, or heck, go to Baja and save even more money (food is super cheap there) and temps are great.
Partly Summer in the Flagstaff - nearby a big city, better temps, but you wont last too long before rangers catch up.
End of summer in high country of Colorado - tons of different public lands that allow 2 week stays, all at 8000 ft or higher giving you perfect temps
Fall in the Zion Ntnl Park area is amazingly beautiful with numerous spots to relocate regularly too, and all close to towns
Beginning of winter just outside of Las Vegas on Lake Mead, because how cool is it to be in Vegas on new years?
Then back to SoCal or AZ for Jan-March again.
The above route is about 1800 miles round trip and even at 12mpg is only 150 gallons ($400-500) of fuel... for the entire year. You can be near enough to towns to easily resupply without it being a huge cost in additional fuel. These areas are regular spots for many full timers, so you will meet new folks and the Rangers are accustomed to this lifestyle.
The other answer (which many have already said) is to find private land to camp on. Ideally it would be someone you know and knows you and is willing to let you camp there for free, or trade for some type of work. Of course, we all want that don't we? Not a lot of that happening either. It's a dream... a fantasy, that only comes true for very, very few.
