No, they have incredible amount of data, no one will store it
Not every nodes have to store everything, only the index and public key/signature of the data. Each node can store locally only the data it needs.
But i don't think either blockchain are really going to be competitive for mass free social networking, more for archiving or small to medium user base, for less than 100k posting users or so it can be manageable.
Or it needs some sharding, but that could be easier to do on applications or social networking that already have information on circles and groups, like data for a group would be very easy to shard, and not everyone has to store all the data from the all groups, while data integrity as well as origin can still be checked publicly.
But at the scale of facebook or youtube, handling millions or billions or user seems difficult, or with a very efficient sharding, because all together most user only need to access a very small amount of the whole data, and data relationship/hierarchy can be made easy to shard, as the relationship of new content with existing content can be made easy to figure out in lot of cases, unlike transaction where any transaction can potentially reference any other transactions, content on platform like facebook can still be segregated more easily.
But the advantage is also not huge, because what make blockchain interesting is in the way it can settle disagreement on transactions which can include ordering or dependency problems, with files or social media there is not a huge need for conflict resolving and it's easy to end up with the same data on all nodes, as there can hardly be some mutually exclusive branch, unless it rely on access on limited resources like transactions depends on coins, there is not much at stake with the consensus on regular application data like for social media, the advantages would be content signature and timestamp, making the application easy to distribute in a secure manner with the cryptography, making it censor free on the global chain even if all nodes can filter the data they want to have locally and/or serve, and making page generation and data sourcing publicly verifiable.
Social media also generate lot of waste and temporary data that is only going to get viewed for short period of time, like instagram or instant chat, so for casual non important temporary data, blockchain are not necessarily a good target, but for a target like archiving, or blog post that are supposed to stay and be viewed for a long time it could have an interest.