Maybe; realistically, the Internet speed would probably be more of a limiting factor. At mine, it would take almost a week to fill up 1 terabyte of storage. That's plenty of time to go get a few more HDDs. At 1 Gbps you'd have about 2 hours.
According to http://www.datacenterknowledge.com/archives/2013/01/18/faceb... , people upload 350 million new photos per day, and they have 240 billion photos total. If we use 1 MB per photo as an average, you'd need about 334 terabytes to store a day of Facebook's pictures. You'd need 228882 terabytes to store the whole thing. Assuming Wikipedia's estimate of $35/TB for desktop hard drives, it would cost about $8 million to store Facebook's photo archive (that's without videos or text). Actually hosting it would cost even more.
(228882 TB to Gb) sec to years = 58 yr 3 w 1 day 18 h 40 min
58 years is going to take a long time
That's why this is more a storage problem than a bandwidth problem in practice. That's a lot of hard drives. While it might be possible to build up a botnet of a few hundred thousand slave machines to download and store it all, the task is by no means trivial.