Welcome to iMore! A few things to try:
Plug your iPhone into a Mac or PC running iTunes. If iTunes recognizes your phone, create a backup. If you get a new iPhone you can restore it from that backup and retrieve your stuff. If iTunes won't recognize your phone you may be out of luck unless you have been backing up to iCloud.
Go to
www.icloud.com and log in using your Apple ID and password. You should be able to retrieve a lot of your stuff there.
Also, your emails are most likely stored remotely on your email server unless you have a POP3 service where emails are downloaded locally onto your device and removed from the server. POP3 isn't all that common these days.
Good luck! If you have more questions regarding this issue please join the forum and reply back within this thread.
How to Sign Up at iMore