can you not set yourself up another email address and set up microsoft outlook then you can forward the emails you want to keep into your outlook file (please don't ask me how to set it up ) but i have a ntl/virgin email (limited like yours) and a yahoo one which i can send to
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?