I'm guessing your hold switch might be broken
you can confirm the function of your hold switch by enabling the hidden hardware diagnostic mode:
1) reboot your iPod by pressing and holding the Select(center) button and Menu buttons together for several seconds.
2) when you see the Apple logo, immediately press and hold the Select(center) and Previous(left) buttons together for several seconds. You should hear a beep and screen will show "SRV DIAG BOOT"
3) Press Menu button to enter Manual Test mode
4) use the scroll wheel or Next(right) button to move the cursor to "IO" (I/O tests). Press Select(center)
5) move the cursor to "HeadphoneDetec
6) now slide the Hold switch, you should see the display change to "Hold: 1" followed by "HOLD PASS". If you slide the Hold switch again, you should now see "Hold: 0"
if you DON'T see the hold switch status line change when you flip the switch back and forth, your hold switch must be broken and needs replacing.
if it's not the hold switch that's broken, you can run the separate test for the click wheel - perhaps there's a chance it is broken and activating the pause button when you don't intend? (not as likely, IMO) follow the similar procedure mentioned above, except after selecting the I/O tests, choose "Wheel" then "KeyTest" - you can then verify that each of the click wheel buttons only respond when you press them (check to see that the pause button doesn't activate without your press)
press Menu button twice to get back to the main test menu, then select "Reset" to get out of diagnostic mode
the repair guides for the iPod Classic are here
you can look for the part you need for your particular model of iPod here
I had the same problem and using the debug menu mentioned above, I could confirm that the headphone jack was faulty - it sensed falsely that the headphones had been removed and paused itself.
My first try was to use a small flathead screwdriver to gently nudge out the little contact metal inside the headphone port.
Miracously, that worked and it seems that I got my iPod back to working normally.
So try this if you have the problem: bending out the little piece of metal you can see when you look in the headphone port, using a tiny screwdriver or a needle, paperclip etc. The contact will be permanent and no faulty pausing will occur.