Get yourself a Gmail/googlemail account.
Then all your emails are online & accessible from anywhere and you can store 7gb+ of mail.
I currently have over 77000 emails stored on there & can get access to every single one instantly.
Although how you'd go about importing all your existing mail in - not too sure - perhaps forward them all
Has your tesco mail got export facilities?
Or setup POP3 access and download all your mail into MS Outlook using POP3.
IN fact - looking at Tesco's it appears that they already use OE/POP3 - so how are you accessing your emails?