Just because your iPod Nano refuses to charge doesn't mean your device is on it's last legs. On the contrary, it's likely not a problem with your device so much as it's a problem with what your plugging the device into. Before you write your iPod Nano off, run through some quick-and-dirty solutions to see if you can solve the problem on your own.
Computer is in "Sleep" Mode
If the iPod Nano is connected to a computer and that computer slips into "Sleep" mode, the iPod Nano will stop charging. You must turn off "Sleep" mode before stepping away from the computer if you want to continue charging the iPod Nano. Also, do not close the lid on any laptop an iPod Nano is connected to for charging.
Low-Power USB Port
An iPod Nano must be connected to a high-power USB port, not a low-powered USB port. The USB ports on a computer are high-power USB ports and will charge an iPod Nano without issue. However, it's easy to be tempted to plug your device into available USB ports on telephones, keyboards, monitors or any number of other devices, but often those devices are low-power USB ports that don't put out enough power to charge an iPod Nano.
Faulty Cable or Port
If the iPod Nano appears to be working properly but doesn't charge, try connecting it to other computers to charge it and also try using a different iPod USB cable. USB ports in computers can sometimes become defective. Connect other devices to the USB port to see if it recognizes those devices. Cables can also become worn or damaged over time and may need to be replaced.
Perform a soft reset on the iPod Nano if it is locked up or frozen, as a device in this state won't charge. Hold down the "Sleep/Wake" button on the upper-right edge of the device and the "-" button on the upper-left edge of the device for eight seconds, or until the gray Apple logo appears on the screen. After the device resets and reboots to the iPod Nano home screen, attempt to charge the device again.
Some car chargers are only built to work with specific iPod models and will not play or charge the iPod Nano. Also, the iPod Nano can be plugged into some car adapters so music from the device can be played through the vehicle's speakers, however, not all adapters that play an iPod Nano also charge the device. Read the box carefully on any iPod Nano car adapter or charger to make sure it's both compatible with your device and will charge it — not just pump the music through the car's speakers.