Assuming you leave them on the server when you access them on the phone, your best bet is to also use a computer to access the whole lot through an app which will copy them to local folders (assuming you want to do what I think). But it might be worth selecting a couple of mails and seeing whether your device's email app has an option to export or copy the mails to a local folder.
You won't be able to directly access the location the device currently stores them unless you root the device. Android apps usually store their data in /data/data/<application name>, most likely in an sql database file in the /databases subdirectory. On an unrooted device you won't be able to access that directory to copy the files, and they may well not be stored in a format your other application can read.