They also didn't have much in the way of worthy competition. That really helped them out a lot. It's different now. So much saturation in many areas things have moved from the era of bold ideas to the era of carving out niches.
Facebook (launch Feb 2004) was founded after Myspace (launch Aug 2003), which was founded after Friendster (2002). So there was certainly competition.
Was it 'worthy competition'? Hard to say without hindsight bias (only the winners 10 years later were 'worthy'), but I doubt Mark Zuckerberg or the Google founders felt there was no worthy competition. It seems fairly clear to me that Facebook (and Google) succeeded because they made better decisions (on average) at a gazillion small decision points, not because they had some great idea that no-one else had thought of.
I remember seeing a talk by Zuckerberg (or another early Facebook person) where they talked about clones of Facebook, and how they were racing to win over different colleges / universities before other social networks did.
I remember one particular college-targeted social media site (besides Facebook) at the time called CollegeClub. Met a couple women through that. It didn't have anywhere near the reach that Facebook did a couple years later, though.
Oddly enough, Facebook was founded at almost exactly the same time as a fast-growing social networking site backed by a famous company with global reach. That was Google's Orkut.
From Bessemer's anti-portfolio:
Cowan’s college friend rented her garage to Sergey and Larry for their first year. In 1999 and 2000 she tried to introduce Cowan to “these two really smart Stanford students writing a search engine”. Students? A new search engine? In the most important moment ever for Bessemer’s anti-portfolio, Cowan asked her, “How can I get out of this house without going anywhere near your garage?”
Wrong on both counts. Both companies entered markets considered completely saturated.
Friendster and Myspace were the big players for Facebook, Lycos, AltaVista, and HotBot were competitors for Google. A new social network or search engine didn't get noticed back then.