It really, really sounds like the problem is your pc, not steam.
It could be a PC issue, but I get the same problem on my ultrabook too. Now, that might not mean much, because the set-up there isn't vastly different to my desktop, and they use the same modem and line. But in both cases, all my other internet connections/transactions (which do include stuff a little more complicated than games sites) are OK, so I'm not so sure... unless you have a specific idea in mind that makes you go "really, really"?
By the steam store website, I assume you mean store.steampowered.com? Now, if only I could connect there...
I don't seem to be able to connect to buy stuff on Steam whether through the Steam interface or through the website.