Try using another web browser like Camino or Safari (which is pre-installed) to replicate this strange issue.
Try using Software Update in your Macs Admin account. If you are already using an Admin acct for everyday use, this is a bad idea and it's strongly suggested you create a Standard User acct and try to download something there. This will tell whether or not the issue is System Wide or User Specific.
Checking the User Acct Parental Controls is worth checking as per Kavemans advice.
Do you by any chance have LittleSnitch installed and running? This app is capable of blocking access to the web but you haven't stated whether or not you have unrestricted access to the web ...