You need to get that repaired. If you asked me a couple months ago, I would have fixed it since I've been repairing Apple products for about 2 years now. If you want to take the journey on your own, just know that iPods are significantly more of a pain than iPhones (at least for me). You can use the touch assist in your settings though so that you can use the home button, but it will be from your touch screen, not the actual button itself.

To get the Home Button, navigate through the settings as follows:

Open Settings >> General >> Accessibility >> AssistiveTouch >> "On"

Is the button just not depressing or is it depressing but not functioning?

If its the first it needs a new dimple on top of the homebutton mechanism, if its the latter then it will need an entire homebutton flex which is a little more difficult to repair.
Either way you'd be better off taking it to a repair store to fix it, its not a good idea for anyone to work on an iTouch without much experience.

-Mattchew wrote You need to get that repaired. If you asked me a couple months ago, I would have fixed it since I've been repairing Apple products for about 2 years now. If you want to take the journey on your own, just know that iPods are significantly more of a pain than iPhones (at least for me). You can use the touch assist in your settings though so that you can use the home button, but it will be from your touch screen, not the actual button itself.

To get the Home Button, navigate through the settings as follows:

Open Settings >> General >> Accessibility >> AssistiveTouch >> "On"

Thank you, now i dont have to turn my ipod on and off again when changing apps. xD