For some reason, Internet Explorer and Chrome were unable to connect to any web sites, while Firefox was able to. It seems that there was a failure in WinHttp as well. I was able to ping www.google.com and www.microsoft.com.

Initial reboot of the computer didn't work as well. I then ran Fiddler, and when I used IE to browse www.google.com, nothing showed up on Fiddler, even though I've updated the proxy to point to Fiddler's 127.0.0.1:8888.

Then, I enabled both 32-bit and 64-bit WinHTTP logging like so:

netsh winhttp set tracing trace-file-prefix="C:\Temp\WinHttp" level=verbose format=ansi state=enabled

and finally disabled it as follows:

netsh winhttp set tracing state=disabled

After sufficient tracing, I found out that the winsock layer needed a reset:

netsh winsock reset

followed by a reboot of the computer, and then everything started working.