On my home PC, everything works fine.
On the PC I'm using at the moment, I log on, I get the "Connecting to Steam account..." message & then nothing.
The box closes.
I'm not logged in.
I try again & get the same response.

Any clues?

Thanks for your help.

Maybe you left steam on at home and that causes problem. =)

Or, you could install steam again. With origin I had some trouble to get online after some update. Re-installing it worked.

Or, firewall is interfering. Sandbox or something.

Well, Steam is online, so no meteor strike got the network.

Could it be a firewall or blocked ports? Looks like that.

Is it at work or anywhere where people might restrict your internet?

I am not logged in at home.
IF I was, I would expect the "You are already logged in. Do you want to override?" popup.

It is listed as allowed in the Firewall.
I am not at work. I wouldn't even try...!

I will try re-installing Steam.

OK, that didn't work.
Error 1316: Cannot read from steaminstall.msi

https://support.steampowered.com/kb_article.php?ref=7247-OPVB-9949

Maybe that helps.

Try changing your region (Steam --> Settings --> Downloads + Cloud --> Download Region).
That worked for me once when I was downloading a game, and once when I was updating a game.

EDIT:
Sorry, I didn't read well your post. I thought you could log in into Steam, but not connect online.

Maybe the Steam client is not properly installed. One way to fix it, if that is the case, is to go to where Steam is installed, and delete everything except for the folder "steamapps" and the file "Steam.exe" and then to load "Steam.exe". The download/update process should start. Is the same process as to transfer the Steam folder to other place/drive, except you're not moving it elsewhere.

Just tried this.
Got the Steamguard code required message.
Checked my email & input the code.
Then again nothing.

The computer you're using now is public (cybercafe, library, university...) or private in your job?. I know this will sound as a cliché, but maybe a firewall or some security software is blocking the Steam client.

Try the link provided by Antimateria, it should work.

If not, go into Steam's root folder (by default: C:-Program Files (x86)-Steam) and delete a file called "ClientRegistry.blob" and then attempt to restart Steam.

It's my mum's computer.
Firewall is set to allow steam.
It's never had a problem with MSE blocking it before.

The link provided by Antimateria has some useful information and tips to fix that error:
https://support.steampowered.com/kb_article.php?ref=7247-OPVB-9949

Failed again.
The autofix didn't fix it.
Did it manually.
Now steam appears to re-install correctly, but back to the same problem.

Sign in, get "connect to account" message & then steam closes itself.

Do you have installed that latest Windows Installer in that system, as Steam suggests?
http://www.microsoft.com/en-us/download/details.aspx?id=8483

And the latest version of Steam?

Duh!
When I read that, I assumed it meant the latest Steam package.
As I downloaded it fresh this morning, I assumed I did!
Trying that now.....

Still no luck.

Oh well, no Red Faction or Children of the Nile for this afternoon.
Back to UFO Aftershock then.