Battery and warning sign

If it's under warranty, you can request battery replacement here.

Service request.

If it isn't, then Apple's out of warranty replacement scheme is a lot more expensive than many of the 3rd party options that are available online.

If you Google "iPod batteries", you should get quite a few hits.

Maybe it's just "stuck." you could try to do a Rest.

Instructions for Reset from the Apple site:
Toggle the Hold switch on and off. (Slide it to Hold, then turn it off again.)
Press and hold the Menu and Center (Select) buttons simultaneously until the Apple logo appears, about 6 to 10 seconds. You may need to repeat this step.

Tip: If you are having difficulty resetting your iPod, set it on a flat surface. Make sure the finger pressing the Select button is not touching any part of the click wheel. Also make sure that you are pressing the Menu button toward the outside of the click wheel, and not near the center.

Other than that, I'm out of my league. Post back after trying a Reset. If that doesn't bring it back, perhaps someone else can help.

In my view it is worth owning a charger. Sometimes if the battery gets completely discharged, simply connecting to a USB port on the computer doesn't supply enough power to "kick start" the charging process. In theory it should do, but yours seems to be a common problem.

You can get chargers from the Apple store or there are many 3rd party brands available form other places.

Perhaps the battery is dead and the charger isn't functioning. Is this a new charger or an old one? Have you tried charging through any other means?