The iPod Nano automatically goes into sleep mode if not in use and that is considered to be “Off” for this particular iPod. If that isn’t working though you can plug it back in and make sure it’s fully charged so it operates properly. If that still doesn’t make the iPod Nano go into sleep mode, you can “force” it off by holding the volume and sleep buttons down at the same time. This way, the screen will go black and the Apple logo will appear. Continue to hold the buttons down. After about twenty seconds or so a list of options will appear in yellow text. The first option is “Standby” and you can choose that by pressing the sleep button. You can then wake it back up by pressing the sleep button again.
If choosing standby still doesn’t work to reset your Nano and help it get into sleep mode, you might need to restore it from your iTunes account. For this you will need to connect your iPod Nano to your computer and walk through the iTunes prompts to reset your Nano. Remember though, the iPod Nano doesn’t really turn “off” as it’s not designed that way. It simply either goes into sleep mode or awake mode.
Resource: Apple Discussion