I need a really quick and easy favour from one or more of you: could you please open the steam website and tell me what you see? And/or open the steam client and try to take a look at the community, store and badges tab and again tell me what do you see? Because I'm unable to open anything steam related (apart from games) and I'd like to know if the problem is on my or their side!

All those tabs are working through the steam client for me.

Working fine for me. If I understand correctly Steam client uses a built-in Internet Explorer browser. Use something like CCleaner to clean out all your internet explorer history/cookies/cache/etc.

It's absurd, I not only cannot access the tabs in the client, due to an apparent lack of connection, but I cannot access the website from firefox or IE either... Very strange, it's the only website not working right now!
And I tried to update the client, so now it won't start at all because it stops updating for lack of connection...

Any suggestion?

It must be an issue with your internet connection or the ISP.

Maybe the server you are connecting to? Where are you as the two who replied are in the States. I am in Spain but it is working fine for me.

I'm in Italy! I used the UK and Italy servers, before trying to update, but it was all the same: connection problem...

And I've CCcleaned everything, but to no avail...

Is there a way to reinstall the steam client without deleting the games from my computer?

Yes.

http://www.youtube.com/watch?v=n5Jp0NF2AKs

http://steamcommunity.com/discussions/forum/1/864958451244951708/

Did you recently update or turn on a firewall? Are you sure your connected to the right wireless AP, if your going wireless.

I'm on Win7, and the last update was last month, so I'm quite sure it's not the case. As for the wireless, I've got a router, and I can't connect to the website with my computer (plugged with an ethernet cable to the router) nor with my cellphone (through Wi-Fi)...

Aaaaand, the second link doesn't work for me because I'm locked out (can't literally load) of every steam site...

Aaaaand, I tried the method shown in the video, and it works, but then steam cannot connect so I'm still locked out...