I don't think we have the technology today make a "global mesh network". We need new Wi-Fi technologies and products to appear, and I think they will over the next 10 years or so. We need to get the Wi-Fi Alliance to create new Wi-Fi standards for longer ranges, too, which would make this easier. It would also help commercial companies, especially if they want to disrupt wireless telcos (such as Google and Microsoft, who have been trying to work on this, too).
But until we get to that point, we can build the technology, and have it more localized in villages, towns, or even big cities. We can use the latest Wi-Fi routers, or mobile phones, to form these meshnets now. Then perhaps we'll get Wi-fi balloons like what Google is building in Africa, although I'm not sure how resilient those would be against a government, but perhaps there will be so many of them that shutting them down would be much like shutting down the Internet.
As for intercontinental meshnets, I think DIY sattellites will become more popular in the future, and could help with this, too.