Dates and times on iOS devices are stored in what's called "Unix time", which is the number of seconds since midnight on 1970/01/01. A date of 1969/12/31 indicates a data field that is empty, implying that the entire email record is corrupt.
First, which mail app are you referring to? Apple Mail, Gmail or other?
Since most email applications use IMAP or some form of centralized cloud storage, they all usually have a web interface as well. I'd start by logging in on the web (for example, gmail.com) to see if your mail looks correct there. If your mail looks correct, the corruption is likely only on your device, which is good. If the mail on the web interface is corrupt, you may be out of luck, unless you're making regular archival backups.
Assuming your luck is good, your best bet may be to remove the account from the mail app on your device (do not remove the app, and do not remove the account from the web interface), which should remove all of the corrupt content (the emails) from your device. Then re-add your account to the mail app, which should repopulate the data.