Five years ago I quit working for other people and started working for myself. At the same time, we decided to start traveling, to get a better idea of other cultures for our products (which are internationalized) and to see if we found a place we liked to stay.
First off, one thing you notice, once you get out of the USA, is how much of a bubble the USA is in. The perspective given by US news sources, for instance, is pretty narrow. Americans tend to think about the same things all the time and have those perceptions reinforced. Just getting out there will broaden your perspective, of the world, of the markets and of your customers.
There are other cultures that support entrepreneurship, and really, the idea that the US is sort of a pinnacle of this is at the least outdated.
The thing to remember, though, is that these other cultures are different and support it in different ways. Some better, some worse than the USA.
During our travels, we've "based our startup" in Venice Italy, Berlin Germany, Amsterdam, The UK, Mexico, and most recently Chile.
All of these are quite viable as a place to stay while working on your startup, though not all of them are good jurisdictions for your business to be domiciled. We travel on tourist visas, though in recent year's we've focused on getting year long visas.
Like others said about the PT / Five Flags strategy, jurisdictional arbitrage is a good idea. One thing to be aware of is that the IRS is getting more and more draconian every year and US government laws are making it harder and harder for financial institutions to do business with americans. It's not that the countries won't welcome you, it's that the US punishes them for it. (At least as it comes to bank accounts.) This makes a foreign bank account a bit of a prized possession for Americans.
Some notable places to consider:
Amsterdam- There is a US-Dutch friendship treaty which means that Americans who found a company in the netherlands get residency so long as they keep a certain amount of assets in the company.
Panama- Recently the president of panama signed an executive order giving anyone from "favored" nations (eg: ones panama has good relations with, including the US and UK) the ability to get residency simply by setting up a company in Panama and putting some money in the bank. This isn't just a residency visa, this is permanent panamanian residency.
Chile- I'm sure you've heard of Startup Chile. Chile is a pro-business culture (though also a bit pro-paperwork and things take a bit more time than they should). The Chilean government will give you residency for living and working in Chile, you just have to earn money in the country and pay taxes on it (So you could go there as a freelancer and live.) After 2 years of doing this you can apply for permanent residency that allows you to live there indefinitely.
Singapore- Has two programs, the Entrepreneur Pass and the Entre Pass for employees of companies (it could be your own company) and people setting up singaporean companies, giving you residency.
All these situations give you a reasonably straightforward path to residency. All of them have more or less pro-business environments, though Amsterdam's taxes are kind of extreme.
As for embracing failure, etc, this is really kind of incumbent on yourself. IF you embrace your failure, great.
If you're looking to raise investor money, the bay area is the probably unparalleled for this. I'm speaking more towards someone who wants to bootstrap.
If you're a bootstrapper, you could do it almost anywhere.