Does the iPod work normally, when you are using it to play songs?
I don't know why it does not go into Disk Mode. You said, "The screen just stays the same." The same as what? The first part of the Disk Mode procedure, pressing and holding down the Menu and Center buttons, is the same as doing a standard a Reset (restart). The Apple logo should appear on the screen (when you do a Reset), so the screen should not stay the same. As soon as the Apple logo appears, press and hold down the Center and Play/Pause buttons, which puts it into Disk Mode,
However, Disk Mode does not have anything to do with storage. You iPod model automatically powers off, if inactive. Once powered off, it does not matter if the iPod was in Disk Mode. You should not put it into storage for an extended period (like a few months or longer). The lithium battery may become unable to hold a charge, if left in a discharged state for a long time. An older battery may not hold its charge for more than a few weeks, before it is completely discharged (even if you don't use the iPod). It would be best if you periodically recharged the battery.