Can you disable the Firewall from Norton 360 and check whether you can access the Google(gmail)?
If not, disable AutoProtect and check the same. This is to verify whether any components from Norton is blocking the google components/web site. If you use any Pop-up/Ad blocker or toolbar, please disable that also. You can also try to boot in Safe Mode with networking and then try to access google. In safe mode there won't be any restrictions in your computer to block google.