Battery is screwed
I highly recommend the following hit from iFixIT and they have good instructions. However, are you sure your battery is dead? See section on 'cannot charge'.
The first thing to realize about this iPod is that you cannot charge by USB. Don't listen to anyone tell you otherwise. You MUST change by Firewire. So, you have two choices. You can either search ebay for a Firewire charger or since you will want to sync your iPod with your Mac, you can follow the last section here for syncing, successfully attaching the iPod to your Mac for syncing will also allow your Mac to change your iPod.
Note the Firewire port, not a USB port.
This is where things are going to cost you a little money. Modern Macs have USB-C ports. A USB-C port is also called a ThunderBolt 3 port. Here is what you need:
1 - Purchase a USB-C/ThunderBolt 3 to ThunderBolt 2 adapter.
Apple Part Number A1790. At time of writing it costs $49 from Apple, less on ebay.
Thunderbolt 3 to Thunderbolt 2
2 - Purchase a ThunderBolt 2 to Firewire 800 Adapter
Apple Part Number A1463. At time of writing it costs $29 from Apple. less on ebay.
ThunderBolt 2 to Firewire 800
3 - Purchase a Firewire 800 to Firewire 400 adapter
Elago Part Number EL-FW-ADAP. At time of writing it costs $9.99 from Amazon
Firewire 800 to Firewire 400
4 - Purchase a Firewire 400 to 30 pin iPod cable.
These are as rare as rocking horse poop. You may already have one. If you have trouble locating one, you might consider buying one of the charging adapters as described above because they should really come complete with this cable.
- Connect the USB-C--TB2 adapter to you Mac.
- Connect the TB2--FW800 adapter to the USB-C---TB2 adapter
- Connect the FW800---FW400 adapter to your (hopefully existing) iPod FW400 to iPod 30 pin iPod cable.
- Connect the FW400--iPod 30 pin cable to your iPod.
If you, like me, purchased the iPod when you were using Windows. Your first step will be to restore the iPod to factory defaults with iTunes.