Can you export your emails in Outlook Web App

Here is the enhancement request. You can drag and drop emails to add them as attachments as a workaround (see this answer). You can download an email like this:

1. Start a new message
2. Click to edit the message in a new window.
3. Select all emails you would like to download.
4. Drag the emails to the new message. This will add them as attachments.
5. Click the down arrow on the attachment and download the email message, or send the message to yourself and then download all attachments.

No you cannot. Outlook Web App is purely meant to send and receive email, but does not have a backup function.

If you want to backup email to an external medium, you will need a client such as Microsoft Outlook that supports this.

So, in order to make a proper backup, configure your account in a mailclient that supports making a backup. Just because OWA does not support it, doesn't mean its not possible. In modern mail clients such as MS Outlook and Thunderbird, just entering your email address and password automatically configures the client using autodiscovery. Not going for a mail client to make a backup and strictly limiting ones self to OWA is not necessary.