The whole idea about the cloud computing is to have your data saved in the online server and access them whenever you needed via Internet, preferably via a browser.
Currently, besides my local archives, most of my emails have copies online. However, couple years ago, especially while I was still in college - Fudan University, most of my emails were saved on desktop computers and they are no longer online. Part of the reason is that online storage space was very limited (20 MB if I remember correctly). But, I did keep copies of them in either Microsoft Outlook format or the mbox format used by pine. Hopefully it would be a fun project to revive all those old emails (estimated 500-1000 MB) into the cloud! Bring ancient history to life!
One thing I am not proud about is that I lost my old emails in the hotmail account (inbox before 5/14/2004, sent before 8/12/2006). I was a heavy hotmail user, especially when I was applying for the graduate school. At that time, hotmail didn't have free pop3 access and the way to access hotmail was through either Microsoft Outlook or web interface. Around 2004 when gmail came out, I tried and loved it since. I stopped using hotmail for about 1-2 months. Later, when I login my hotmail account, all old emails were gone! Very shocking accident and I am still puzzled by how it could happen ...... anyway, because of that, right now, my hotmail is only for msn purpose.