I have an orphaned email account and after a good ride, it seems that my time is up and my account has been locked.
While I'm ok with the fact that it doesn't look like I'm going to be able to carry on using it, is there any way that I can get my emails from the account so I can transfer them to my new account?
It depends on how far down the mailbox deletion route the account is. Webmail access is usually the first to go, followed soon after by email client access.
You could try setting up the account in an email client, using POP access, to download the emails to be held locally.
You will only be able to download emails from your Inbox, any held in other folders will not be available.