Q: I can't start FileVault on my Macbook pro.

Take a look at this article, it should help you Use FileVault to encrypt the startup disk on your Mac - Apple Support

How is your disk formatted?

And how is the disk formatted?

So how is it formatted now?

You need a logical Volume structure on that drive to use FileVault. To create that if not there already, you must boot to an alternate boot drive (NOT including Recovery partition on the same drive) and completely initialize the Physical drive using a recent version of Disk Utility.

By memory, I think that is 10.9 or later.

The Terminal command:

diskutil list

will tell you whether the Logical Volume structures are present.

There are no Logical Volume Group structures shown in your Diskutil list display above.

The simplest way forward would be to make a USB stick Installer/Utilities of a recent version of Mac OS X, such as the one you are running now. The free utility DiskMakerX can make the production of such an 8GB USB stick Installer very straightforward.

Such a USB stick is bootable, and contains a version of Disk Utility. When you initialize the entire Physical drive, the default is to create the required Logical Volume Group structures on it.

HOWEVER, this will Erase everything on the drive, so you must have At least one Trusted Backup, and preferably two using different methods..

That is one Trusted Backup. The most Prudent way to proceed is to make another copy on another drive to protect against a problem with the Time Machine copy, as initializing your boot drive leaves you with ONLY the Time Machine copy, and no other, until you can successfully do the Restore. If something goes wrong, you could lose data.