Mothballs in a cedar chest? Wonder why...

Are you sure it isn't the natural smell of cedar, which can be very strong?

As to refinishing, no way to know. Depends on how bad the existing finish is, and what kind of outcome you are looking for.

If the finish isn't badly damaged, I recommend Formby's facelift. Some years ago I bought a Lane cedar chest from a Good Will for $12. The previous owners used it for a toychest. It had water rings, the works. The Formby's facelift kit did a fabulous job. It doesn't strip the finish. So if that is damaged, you'll need something else.

Also, if it really is mothballs, take it outside and let it sit in the sun (if that is feasible). Air is the best thing to remove that smell.

Good luck!

Airing out and, perhaps, sunlight will reduce the smell of mothballs. Be patient, this may take several days.
Lightly sand the cedar to rejuvinate the natural aroma. Do not put a finish on the inside of a cedar chest.

randy has the correct solution.
You will not get the odor out unless you sand it out.
I will echo exactly what he said. Sand the cedar to rejuvinate the natural aroma and DO NOT finish the inside,leave it Au Natural.

Agree with Randy.
A light sanding with 600 grit wet/dry paper is a good start.

Try googling "cedar oil". This oil is extracted from aromatic cedar.
Wipe it on a few times and it may help the situation

Here is a link that might be useful: Cedar oil from Woodcraft

Why would you use a 600 grit paper and WHY a wet/dry paper.
We're not afraid of the sanding process here.
220 grit by hand or on an orbital sander will be just fine.