Hello Markbmb -
I was in a very similar problem this last month with an Ipod classic fourth generation, 20 gig unit.
I wanted to get it fixed after owning the unit for 11 years and almost that entire time it would never work, so I called Apple about it a few weeks ago.
They directed me to my nearest store and an Apple associate advised me that they do not work on the ipod classic fourth generation models anymore
and that I should try to find a third party company to work on it and I was basically out of luck. So I did some researching online and found a few companies in my area that worked on them, but nobody would touch the model I needed to be fixed. I researched the issue further and after some digging online, I decided to take the unit apart myself and I am proud to say that I was able to repair my ipod on my own. Its a little scary at first, but its actually easy to do. The hardest part of the repair is the very first step - getting the ipod open.
You are going to need to repair this one yourself. I bought my replacement battery off of amazon.com for about around 15 bucks, and had to pay a little bit more in shipping. It should come with a kit with a special screwdriver to unscrew the screws holding the board down inside, and some plastic tools that help with the process to remove the battery. Supposedly they help with separating the ipod shell to open the unit too, but I found them to be less effective than a guitar pick or precision thin object. I found a utube video online to address the process to open it and repair the problem.
I will be happy to describe the steps on the repair process in greater detail if it helps you to go all the way, just let me know. imo Apple should be helping us with this model but I am happy to do it in their place.