A deep web marketplace, like the recently departed Silk Road, is an Ebay of anonymous e-commerce. Other elements of the stack include:
Could you could go down the list of Amazon Web Services and build p2p distributed versions of each service? I don't see why not.
Could you have a distributed version of Heroku/Amazon Web Services, an anonymous, distributed platform with all the services and APIs to create any app or marketplace from Ebay to Uber running in your browser or on your phone? I don't see why not.
Could you have a distributed p2p version of UPS like TOR, with people handing each other anonymous packages and delivering in some dead drop or to the holder of dollar bill number B12345678?
Of course, that's what they would do in an underworld network, or in a totalitarian state. Like samizdat publishing.
There's a constant ebb and flow between centralization (mainframes/AOL) and decentralization (PCs/Web) and back (Cloud/Facebook).
Similarly there has been an ebb and flow from relatively anonymous Web protocols like Web, SMTP etc. to trusted IDs, Twitter/Facebook/SSH, and back with Bitcoin and dark web.
There's a dichotomy: A world where much of the communication, transactions, commerce have to be over a dark web would be a pretty effed up place, like one where people had to pass along literature through samizdat and do commerce in back alleys.
And yet, it's insane for someone like David Cameron to say the government needs the keys to everything and there can be no true dark web. If you had to put a back door in every communication or ecommerce system, impossible to believe anything would be safe against black hat hackers and foreign governments. And of course bad guys would always find a way around it. And there's a free speech issue: what kind of world is it where you can't have a private conversation in the safety of your own device? It's unachievable, dangerous to try, and wrong.
Every time you use control of legit platforms to achieve political goals, whether it's against Falun Gong or Russia, you create demand for the dark web.
If you don't want people to use the dark net, don't mess up the legit networks with back doors and warrantless wiretaps, 'express lanes,' censorship, using them for political pressure.
Or people will create worse versions and route around you.
1 Technically, these aren't distributed in the same sense as the blockchain is. Many distributed apps and use cases could probably use them as a storage layer, though. For instance, to build a distributed p2p Uber, drivers could have an app that posts availability and reservation responses to a Pastebin type public space signed with their key, and riders could likewise post reservation requests. Perhaps there's an opportunity for a NoSQL nontransactional distributed p2p database counterpart to the blockchain, which is transactional, strictly enforces no double-spend, but takes a long time to commit.