As Biggs said, it sounds like your battery has finally died. It has either been charged too many times (somewhere around 500 times or so they will start to peter out and few make it past 1000 charges) or the battery has sustained massive temperature swings, whatever the reason, you'll need a new battery.
You can do the Apple Authorized way (which is to pay 6.95$ s/h and 59$ parts and labor) The advantages of course are that apple is liable for any damage during the installation of the replacement battery, and the product receives a temporary warranty against battery defects after you receive it. The disadvantage is pretty obvious - that's hella expensive. Plus you are without your unit for a week to a week and a half for shipping and the actual labor on the unit.
http://www.apple.com/support/ipod/service/prices/#us That is the sheet explaining battery pricing
The other option to get the battery replaced - and the one I personally recommend - is as Biggs suggested, that is, purchase a kit

The vast majority of kits will include batteries and all the tools you need, although you may have to do some soldering with this model.
Instructions can be found here: (includes step by step pictures, please for the love of god use a grounding mat or bracelet or you will end up needing to replace the logic board, its super sensitive

)
http://www.ifixit.com/Guide/iPod/iPo...ry/99/6/Page-1
make sure the battery is 400 mAh, 3.07v lithium based. You can either purchase it from ebay or another auction site or through craiglist (cheapest) or through a retailer who normally installs the products for a labor fee (just ask them to send you the battery rather then you sending the ipod and getting the battery installed, ipodjuice.com will do this for you, in fact if you decide to go with them use code "freeship" to remove any shipping charges) Getting it from one of these guys is the method I personally prefer, since you are dealing with a legitimate business and not some possible douche bag.
Let's regroup. You can:
Send it in to an Apple Repair Depot for ~$70
Purchase a Battery from an auction site and install yourself for ~$10+s/h
Purchase a Battery from a 3rd party retailer and install yourself for ~$20+s/h
Purchase a Battery from a 3rd party and have them install it for you for ~$35+s/h
Purchase a new iPod 4th Gen Nano from Apple for ~$150 no s/h
Purchase a refurbished iPod 3rd Gen Nano from Apple for ~$99 no s/h
You'll also want to keep in mind the warranties that come with those options. Typically an auction site will not include a warranty, and a 3rd party retailer will have various warranty - some go up to a year or more. New/Refurb iPods from Apple all come w/ 1 year warranty plus you can extend the support for an additional year with an AppleCare Protection Plan.