More parameters are needed, in my opinion. What you want is possible, but not with only $10k. That, or give up some requirements (internet, unless you want to walk to a public library) and be prepared to squat on someone else's property.
I live close to your desired existence. I own my humble hermit's cabin outright, have no debt at all, and a few grand in savings. I grow some of my own food, hunt some more, but I still need supplies. I could conceivably coast through a few years without any external income, but it would be a very Thoreau-like existence (which isn't a bad thing, just too primitive for me). I own my car and home, but still forced to pay insurance and yearly registration/taxes (cheap, as the car and home are old and of very low value, but they're non-negotiable and will surely rise over the years and are unavoidable). I still pay for power and internet, which could be trimmed (but what's the fun in that?).
If someone mailed me a check for $100k today, I'd never need to work again, and I'm under 40. I'm that set up and frugal. As it stands, I pimp myself out as a remote admin and local PC support dude, with minor hopes that I can realize a small web site that will bring in a few kilobucks a year.
For $10k you can buy an acre on Hawaii ("the big island" -- subdivisions near the lava flows), fly out there, and start gardening year-round while you live in a tent. Water falls from the sky daily and you can raise some meat to supplement the fruit and veggies you can grow. You'd have enough cash to pay for a few years of taxes, too.
Property taxes will get you every time, though, unless you squat. There's some dude who lives in a cave in southern Utah on public land. He eats bugs and scavenges food from nearby towns. Interesting experiment, but he'd be hosed (likely) if the majority of people weren't wasteful and he lacked stuff to scavenge.
Anyway, it's possible, but you'd either need more money up front, to have everything owned outright when you begin, or have enough passive income/savings to cover taxes and minimal supplies. Like I said before, more information on your current situation and your goals/plans are required for a more realistic analysis.