Email apps on your phone can't manage and maintain your actual email account data, they're all just email clients. You need to look into the email server, where your email account resides. Those online servers are what your email provider store your account data.
So which email service do you use? Gmail? MS Outlook? Yahoo Mail?
Log into your account using the web access site of your chosen email service and there's likely to be some kind of export function. With Gmail as an example you can use Sign in - Google Accounts to make a backup of your email/calendar/contacts. Or if you have a third-party application like Thunderbird there's an Export function in the Accounts menu.