Sorry to understand that your email has been compromised; is there anything we can do to assist you with this from our end? Can you advise why you think it has been compromised? Have you managed to change your password at all?
Please do let us know if you need any further assistance from us.
I think we as a community would like to help more but you are being so enigmatic and cagey about giving information that it makes giving help hard. I'll give you a few pointers.
First of all if the comprised account is your VM primary account then it can't be closed, deleted or otherwise got rid of. You and VM are stuck with it for the duration of your time with VM. In fact it is highly undesirable not to try and regain control of it. Firstly to see what damage the supposed compromise might have caused and secondly to get control back by changing the password and security question in My Virginmedia to keep the contents safe.
If you decide that you are going to open another email address with VM then as long as you have secured the compromised account you will still be able to access it and view your emails. If you need to action them you can always forward them to to your new address. Better still if you set up unified email in VM web mail you will be able to see the contents of both addresses at the same time and move between them with ease.
As for downloading what apcyberax meant was , I think, setting up an email program like Thunderbird (called an email client) and downloading to that. You could always look at a product like Mailstore and use that as a secure back-up of that mail.
But your priority should be secure access to the compromised account then you won't have further issues.
When all this started I was being inundated Norton mainly and McAfee emails presumably fake which just changed addresses to get round filters. VM didn’t seem to recognise them coming in but wouldn’t let me send a copy out. Eventually my account was blocked and contact with VM was virtually impossible, also ended up with another on the net when trying to access saying Forbidden 403 error. All this over a period of about 6 weeks, until I managed to get someone on chat at VM who cleared the block. My AV provider helped me clear the 403 error. Since I found and deleted the 100 or so unwanted addresses in my collected contacts spam and trash seem to have dropped to a trickle - they never should have come in anyway, none were read! It is now manageable especially since I have up my Virgin Mail in both Outlook and W Mail.
So I think we can leave it there for now. But thanks for your efforts.