Well for this thing you have to go through two methods:
1) Rooting your device
2) Backup your email [Need Root].
For Rooting Your Device:
1) Download Framaroot app from Play Store and Install it on your device.
2) Open the app -> In Specify Execute type-> Install Super SU.
3) You will see some exploit character will appeared, click any of the exploit.
4) You will see a message popup- "Success, SuperSu and su binary is install". Reboot your device. That's all your device is rooted.
Source : http://menonjats.com/root-or-unroot-android-using-framaroot/
OR incase above method not work.
1) Search "Root Genius Software for PC", Download this software and Install it on your PC.
2) Connect your phone with PC using USB Debugging.
If you don't know how to open USB debugging below link will help you :
3) Open Root Genius Software and Press - Root.
Source : http://androidxda.com/download-root-genius
For Email Backup :
You can use adb for this:
adb backup -noapk -noshared -f gmail.ab com.google.android.gm
would backup all data of the GMail app to your PC, storing it in a file named gmail.ab in the current working directory. A restore of that snapshot then is as easy as running
adb restore gmail.ab
Note that this will include all the settings of the GMail app.
If you prefer doing the same via an app on your device, take a look at Helium Backup.
Without root, it's not possible to access the corresponding folder on your device directly (which would be found below
/data/data/com.google.android.gm and be owned by the GMail app, which forbids access for other apps/users, including the shell user).
Original Source : Backup gmail emails offline