I always thought that "web3" meant everything was on the blockchain. In other words, you could theoretically re-invent Twitter as an Ethereum Smart Contract.
In doing so, every "Tweet" would be on the Eth block chain. You'd probably want a web page that interacts with the block chain to read/post tweets, but even a static HTML file on your local hard drive that interacts with Metamask could achieve that. You wouldn't need to run a site.
Of course, therein lies a new problem: Scale. A quick Google says that there are 6,000 tweets sent per second. Meanwhile, the Eth block chain handles what, 10-20 transactions per second, and ends up with fees that are several dollars worth of Eth? Nobody other than political figures and spammers are going to want to pay $5 to send a Tweet.