I have been through six Xbox 360's since launch in 2005 and I have had four instances of the RROD. The above answers are correct in that it is due to hardware failure and has NOTHING AT ALL to do with the Internet. It usually has to do with overheating. Microsoft carries a 3-year warranty for RROD, and they will fix or replace it for no charge, including them paying for shipping (I've done it four times). However, as poohbowen mentioned, there are guides online that can help you fix it yourself if you are out of the three-year warranty. BTW, as time has gone on since they launched in 2005, Microsoft has continually upgraded different pieces and parts within the 360's to try to prevent these failures, among other things. All Xbxo 360's are not created equal. And FWIW, from my experience each 360 I have gotten through the years has been more and more reliable, as well as running quieter and cooler. My newest one runs great (so far....crossing fingers & knocking on wood ;-)
Whether or not the Xbox 360 has been connected to the internet is irrelevant. It can get the RROD ("red ring of death") either way. It's caused by a hardware failure. It's not anything like a virus that you can accidentally or unwittingly download.
Yes the red ring of death can happen even if it hasn't been connected to the internet. The causes are insufficient power and overheating. There are some things you can do before sending it back to have it fixed. Check out this site for help.