I had a very similar problem. The charge on my 4th Gen Photo battery would last less than 30 min. before completely depleted. In addition, I would plug in the firewire charger into a wall outlet noting that the date and time would reset to Jan 01, 2000 0000hrs. I did not lose any on my tunes. Thus I only listened to it if I was able to plug in in to an outlet somewhere.
After seeing the replace battery suggestion noted above, I did just that. I disassembled my iPod and discovered that the battery within was bulging from the center --- (I had a similar problem with a cell phone battery that fail to maintain a charge) --- swapped out the batteries, charged it up and have not had a problem since. With the proper tools I had purchased here at iFixit (yes, you need the spudger to remove the old battery), my total repair time was less than one hour.