iTunes can't backup iPhone because of an unexpected error.

From the Finder, choose Go > Go to folder. Type /var/db/lockdown and press Return. Choose View > as Icons. The Finder window should show one or more files with alphanumeric file names. In the Finder, choose Edit > Select all. Choose File > Move to Trash. You may need to enter an administrator password. Delete the files in the Lockdown folder but don't delete the Lockdown folder.

Restart, launch iTunes, try now.

freemxy wrote:

Is it certain that I can't cause damage by deleting the files in the lockdown folder

Yes.

freemxy wrote:

When I want to back up my iPhone with iTunes on my Mac iTunes tells me that a backup is not possible because of an error, no other explanation.
I've restarted my Mac and my iPhone, my iTunes is updated and I've tried it several times. I can't backup to iCloud because there are 45 GB on my phone and I don't have so much iCloud storage. It should however work through iTunes cause I have plenty of space on my external hard drive on which all my iTunes data are saved.
I'm thankful for any help.
Cheers.

Although you are using an external drive for all your iTunes data, the iPhone backups still get written to this location, which is on your Mac's primary drive:

<your-username>/Library/Application Support/MobileSync/Backup

Hence you need enough free disk space on your Mac to write the 45 GB. Is there enough space for that?