Thanks for the replies, guys. Your pieces of advice are very helpful.
I am quite aware that one can't compete with Facebook, but one can dream of reaching the popularity of other social networks, such as Netlog, PerfSpot and Hi5 ...
To start with a VPS? Hmm... Seems like a bit too little. I was chatting with some colleagues and we came to the conclusion that - if a brand-new social network, with a really professional design and a user-friendly interface, popped up overnight, there would be a large number of visitors (not necessarily users) who would be curious to check it out. That would trigger a lot of traffic, and maybe a VPS couldn't handle it. A dedicated server, or perhaps even two, would be best, in my opinion...
