Centy, the senders e-mail address is legitimate and possibly just marketing, as a Skywards member I get them all the time from Emirates and they are not too copious and sometimes they are actually quite useful.
The common e-mail transport protocol will often resend failed sent messages with an increasing time gap until it finally gives up. The message you have displayed in your original post is not from Emirates but from your e-mail provider and is telling you your 10Gb mailbox quota limit is nearly full and cannot accept an e-mail sent from Emirates, or from anyone else for that matter until you free up some space.
An e-mail mailbox is comprised of an inbox, an outbox, sent items, deleted items, and any other personal folders that you create to organise your e-mails. If all the messages in all these "boxes" add up to 10Gb in size then your e-mail mailbox will hit the quota limit imposed on you by your e-mail provider. Your e-mail provider may be different to your internet provider e.g. my e-mail provider is Google who offer me a 15Gb quota limit and my internet provider is Optus. Sometimes they can be the same, e.g Telstra (internet provider) have their own e-mail provider brand called BigPond. Outlook is not an e-mail provider, it's an e-mail reader.
Dealing with Telstra has always been a pain, but before you do the standard Telstra ritual of banging your head against the wall, have a look at all the sub folders in your Outlook mailbox then try and clean out as much of it as you can. Outlook does have a few technical ways of determining how much mail you are using but it's beyond the scope of this response. Should that not resolve the issue then contact your e-mail provider for assistance, not your internet provider.
Good luck, and hope you can get it sorted soon.