Skynet is a new protocol that has been working on something like this. You can see some of the webapps built on Skynet here: https://skynethub.io/_B19P18DEI4Y_37wg6yXtfulEq2U8AcIO_lWM7s...
Main project website: https://siasky.net
Main project website: https://siasky.net
In typical p2p networks, you are expected to be both in roughly equal proportions, which means your network gets polluted by a lot of low-quality users who are just trying to do their part.
Sia has the consumers pay the producers, and has a marketplace mechanism that selects the highest performing producers to perform the jobs and receive the revenue.
This allows the network as a whole to be much, much more efficient.
That said, I think there's definitely a space for paid internet. I was just surprised because siasky homepage says "Build a Free Internet" - perhaps it should say "Build a pay per view internet" instead?
I would argue that the current model is more friendly to attention grabbing content (clickbait, etc) because advertising has the limitation that all views are worth the same value. In a pay-as-you-go Internet, users can whitelist high quality content sources as being okay to charge 10x or 100x what you'd typical accept to view a webpage. This would incentive content creators to build a brand and reputation that makes users comfortable putting them on the 'high quality' list, so that their content can see a massive revenue multiple relative to the number of eyeballs.
There are certainly a lot of those, but they are not nearly the whole internet. A lot of interesting sites would not want to be on the "pay as you go model" at all!
For example, we are on the news.ycombinator.com in the thread discussing datprotocol.com, and you are pointing me to siasky.net. Right now, the top 5 HN pages are adecentralizedworld.com, handsonscala.com, arxiv.org, a16z.net, and torproject.com. None of those websites make money from webpage ads. None of them are likely to move to pay-as-you-go internet -- because they care about their audience and not website revenue.
I suspect that Sia's decentralized pay-as-you-go world will be much worse (productivity-wise and information-wise) than the current internet -- all the interesting technical/science blogs and docs would be missing; while buzzfeed clones will be plentiful. There might be occasional high-quality journalism website, but most of those are getting paywalls anyway, so will they justify all new protocol?
Also, the internet is already something that users have to pay for. Users pay their ISP every month for access. We envision a world where this utility payment extends to cover the content creators in addition to the infrastructure providers.
If you want to download skynet files using the command line, you can run 'siad' in portal mode as a daemon, and then use 'siac' to query the daemon. 'siac skynet download [skylink] [destination]' is the command to perform the download.
Doing it this way means you gain direct decentralized access to Skynet instead of needing to funnel through a portal.