With the unlimited storage that Gmail and Outlook offer now, you might be tempted never to delete your emails. But that's not how you get to Inbox Zero. For that, you need a change in mindset. A tidbit from Merlin Mann, who invented the Inbox Zero practice, might be the shift in approach that you need:
Stop thinking of emails like precious family heirlooms, and start treating 'em like pints of milk. Perishable, time-stamped milk that becomes a little less fresh every day until it smells kind of funny and just needs to be dumped. Believe me, there will always be more coming.
Mann rightly says that the relevance of an email is directly proportional to its recency. The older an email gets, the less important it becomes. So there's usually not much reason for you to save them. At best, you can archive such mails and move on. In short, learn to detach from your digital heirlooms and manage your mind space instead.