Apple - Your iPhone could not be activated because the activation server is temporarily unavailable

The two major causes of this are:

1. Network related - change the network used to connect
2. iOS related - something is amiss with the code signing / provisioning

The most likely cause of software is a beta. Apple disables the iOS betas soon after the release happens. If you never manually installed the official release, this can happen. If not, consider backing up your device after checking a couple networks and reinstalling.

If you need to get a beta off, do the following to fix it:

1. Hold the Power and Home buttons until the phone reboots and you see the Apple logo.
2. Release the Power button but keep holding the Home button.
3. Release the Home button when you see the iTunes logo and the connector below it.

Do this while connected to iTunes. Click Restore. Hopefully you have a recent back-up.

Note: It is a large update and will take a while. Apple servers are currently getting slammed.

I have done some research after experiencing the same issue (iPhone 4), however there has been mixed reviews.

What Apple suggests:

This Apple thread on troubleshooting activation issues gives you a couple of options to try:

1. Restart the iPhone
2. Try another means of reaching the activation server (over wi-fi or connecting to iTunes)
3. Restore iOS software - but see my first point under "What I have tried," which leads me to try the next point,
4. Perform a recovery-mode restore (factory reset)

EDIT

5. Some people have reported that if you wait long enough and keep trying to activate, it will eventually go through.
6. Other people have reported that you need to download the latest version of iOS (7.0.2) - namely from the Apple iOS Dev Center. Note that you need an Apple Developer account to see both these links.

What I have tried:

This Apple support thread says you need to restore the phone to a previous backup, from a fresh factory reset. When connecting my iPhone 4 to iTunes, I get this message:

I've tried to update my iPhone (it was up to date, but just in case as I put the first beta .IPA on this phone), but I get this message:

I've tried to restore my iPhone by clicking on the Restore iPhone button, but I get this message instead:

What I am going to try (which will work if you have a backup) is to factory reset the iPhone and restore to a previous backup (see point 4 in the first section).

Some people say to keep retrying as Apple's servers are currently being hammered with requests. It may be worth to wait an hour or so to see if the activation lets you through after a while (if Apple's servers are really the issue).

I found a tip for you. Try to activate your device via Wi-Fi. Just connect to the Wi-Fi (Don't connect to iTunes) and start the activation steps.

Source: http://www.unlockboot.com/2015/10/fix-iphone-activation-error-after.html