Grey screen after macOS 10.13.4 Update

That means you have no Recovery HD. You will need an installer for either the OS that was pre-installed when the computer was new, 10.6.1 or 10.6.2, or the retail 10.6.3 DVD that you can purchase Snow Leopard through the Apple Store: Mac OS X 10.6 Snow Leopard - Apple Store (U.S.). The price is $19.99 plus tax. You will be sent physical media by mail after placing your order. After you install Snow Leopard you will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store.

Snow Leopard Erase and Install

1. Insert Snow Leopard DVD into the optical drive and restart the computer.
2. Immediately upon hearing the chime hold down the C key.
3. Release the key when the Apple logo appears and wait for the loading to finish.
4. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities' menu. After Disk Utility loads select the out-dented disk entry from the side list (mfgr.'s ID and drive size.) Click on the Partition tab in the Disk Utility’s right window. Set the number of partitions to one (1) from the drop down menu. Click on the Options button and select GUID, click OK button, set the format type to MacOS Extended, Journaled. Finally, click on the Apply button.
5. After formatting has finished quit Disk Utility. Continue with the OS X installation and follow the directions.
6. When the installation has finished the computer will restart into the Setup Assistant. After you complete the Setup Assistant you will be running a fresh installation of Snow Leopard. Open Software Update and install the recommended updates.
7. Download and install Mac OS X 10.6.8 Update Combo v1.1.

Then either the disk is bad, the optical drive isn't working, or there is an additional problem such as a dead disk drive or motherboard malfunction. I would try running the AHT:

Intel-based Macs- Using Apple Hardware Test

upekkha-AppleHardwareTest · GitHub

Your machine cannot use Network Recovery. You will need to erase the drive then install Snow Leopard from the DVDs that came with it or the retail copy of the Snow Leopard installer for 10.6.3. Then update that to 10.6.8.

If you haven't erased the drive then you can try reinstalling macOS using the Recovery HD if you have one and it's working:

Install El Capitan or Later from Scratch

If possible backup your files.

1. Restart the computer. Immediately after the chime hold down the CommandandRkeys until the Apple logo appears. When the Utility Menu appears:
2. Select Disk Utility from the Utility Menu and click on Continue button.
3. When Disk Utility loads select the volume (indented entry, usually Macintosh HD) from the Device list.
4. Click on the Erase icon in Disk Utility's main window. A panel will drop down.
5. Set the Format type to APFS (SSDs only) or Mac OS Extended (Journaled.)
6. Click on the Apply button, then wait for the Done button to activate and click on it.
7. Quit Disk Utility and return to the Utility Menu.
8. Select Install OS X and click on the Continue button.

This will install the version of OS X you had installed.

I'm able to run Diagnostics (Command D) and all systems are go. Everything is just fine. There are no hardware issues. I just recently had a full update, verifying everything as healthy.

I'm able to enter single user mode (Command S) volume and other checks are all clear.

I'm able to enter Verbose mode but have no idea how to proceed from there.

But I'm having the exact same issue. It all started when my late 2011 Macbook Pro autoupdated to 10.13.4 overnight. It woke me up rebooting over and over. I haven't been able to do anything since. Nothing else will function.

I can't enter recovery mode, safe mode, or proceed to the login screen.

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The update isn't the cause. This is hardware failure.