Several users have reported difficulty connecting to the game server due to some Steam API/linking issues.[Spoiler]
(I am not an expert in the area, but...) It seems unlikely to be a client-side software issue since both Windows and Linux users have reported the problem and client update/file integrity check revealed nothing, it seems unlikely to be a (straightforward) DNS issue since the users can still connect to website, and it seems unlikely to be a server-side issue since this affects a relatively small proportion of users. My best guess is that there is a problem with some Steam server somewhere.
If you are encountering this issue contact an admin and get them to change your password - this seems to work around the problem.