How does one become eligible to travel free?
Sadly not possible. You could volunteer or work your way around the world.
If your willing to commit to 1 or 2 year teaching contract in china they will fly you there. http://daveseslcafe.com
Generally you need some money to start off or for emergencies. There are foreigners begging in developing countries, not a pretty site.
The smartest thing you can do is to save before you leave, if you could save $USD6000 you could live in a developing country for a year or more not having to work.
I don't know if it's a universal thing, but airline staff often have the ability to travel either for free or for very cheap, so long as they fulfil certain criteria in their travel plans (and are happy to give up their seats to someone paying the full airfare instead).
There are other jobs which pay at least some of their employees' travel costs if the travel is for work purposes. A friend of mine, for example, needs to meet with donors to the university he works for (and that I study at), some of whom are overseas. Travel like that gets paid for - admittedly only in economy class - but then any travel for pleasure is on his own coin.