I use a Synology NAS for my backups.

I have a bunch of scripts that backup parts of my ~/ directory as I see fit (eg photos not music after a photo editing session) to my Synology NAS on an as needed basis.

As such I didn't want to have my NAS running 24/7, mainly because it seems a waste of power but also the lessor concern of wear on the discs (I used IronWolf discs that are built to be running all the time really).

Wake On LAN

Wake On LAN (WOL) is a way of turning on a machine by sending it a special network signal (packet) which causes the BIOS to start the machine booting up. Of course the BIOS needs to be listening to the network so I don't satisfy the ideal of having the machine off completely, but it will be using less power than on an running I assume.

WOL is a feature that must be turned on via the Synology interface.

Login to the web interface of your NAS and navigate to :
Control Panel -> Hardware & Power

There you can activate WOL on the appropriate network interface.

Remotely turning on Synology NAS via WOL

In order to use WOL you need the MAC address of the target network interface.
You can find this through the Synology web interface again:

Control Panel -> Info Centre -> Network

The MAC address is shown there in the format: 00-11-22-33-44-55.
Copy the MAC for the network interface that is connected to the network you will be sending the WOL signal to.

When you use the MAC address you will need to replace - with :
So the example shown above would become: 00:11:22:33:44:55

I use Fedora Linux on my main machine, from their I can use the command below to turn on my Synology NAS using a WOL packet:
$ wol 00:11:22:33:44:55

Of course you will need to replace 00:11:22:33:44:55 with the MAC of your NAS.
If you don't have wol already installed, you can add with your package manager. In Fedora this would be "sudo dnf install wol".

To save myself having to remember the MAC and manually enter it, I created an alias in my ~/.bashrc file:
alias wol-nas='wol 00:11:22:33:44:55'

wol man page