did both sides fail at the same time - without a warning?
it's possible that lint/dust/dirt is inside the headphone jack - but it could also be a faulty headphone jack, the next possible thing could be the logic board - but you have to open the iPod to find out
here is the guide, use it to replace the part - all needed parts/tools are linked listed in the guide.
but at first i would try a reset (press and hold menu and select until the iPod restarts), the would i try to "clean" the headphone jack, you can blow or use canned air to "clean" the jack.
if that won't work - get the tools and open it, use the logic board guide and check the orange parts shown in step 8 - are they still there (and not burnt)?
if they are there, the cleaning was not successful, well - i think the headphone jack is dead and needs to be replaced (omg - what a long complicated text - sorry for that ;-) )