Your SQL server needs an SMTP server to relay your message to the destination email address.

http://mail.google.com/support/bin/answer.py?hl=en&answer=13287 tells you how to configure a client to access your gmail account to read your email.

What you are trying to achieve is to send an email from your SQL server to your gmail account. Most SMTP servers will deny relays from unknown systems. You can blame spammers the world over for that!

You don't need an email account with your company to make Database Mail work, but you do need an SMTP relay server to accept connections from your SQL box.

Is there a mail system at your company? If so, have a polite word with the email administrator and ask him if he would mind adding the IP address of your SQL server to be allowed to use the SMTP relay. If your company has its own mail system, they will have one. If they all use gmail, like you, you may be stuck!