Can I put my phone in "Lost Mode" if I don't have Find My iPhone enabled?

Because there is no way to communicate that option to the device: Find My iPhone: Lock and track your device using Lost Mode

If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support

"If Find My iPhone isn't enabled on your missing device

If you didn't turn on Find My iPhone before your device was lost or stolen, you can't use it to locate your device. But you can use these steps to help protect your data:

1. Change your Apple ID password. By changing your Apple ID password, you can prevent anyone from accessing your iCloud data or using other services (such as iMessage or iTunes) from your missing device.
2. Change the passwords for other internet accounts on your device. This can include email accounts, Facebook, or Twitter.
3. Report your lost or stolen device to local law enforcement. Law enforcement might request the serial number of your device. Find your device serial number.
4. Report your lost or stolen device to your wireless carrier. Your carrier can disable the account, preventing phone calls, texts, and data use.

Find My iPhone is the only way that you can track or locate a lost or missing device. If Find My iPhone isn't enabled on your device before it goes missing, there's no other Apple service that can find, track, or flag your device for you."

No.

It is one of those locking the barn door after the cows have been stolen situations.