This was my grandmother’s cedar chest. It is not a fancy decorative one. It was tucked in the corner of her living room when I was a child and after she remarried at 74(!) and moved, it found a new home on a screened-in porch. The outside of it got a little bit weathered there, and that might be why no one else in the family wanted it.
I have not used it as a cedar chest because it also has the misfortune of smelling strongly of moth balls inside. Everything that spends even a little bit of time in it picks up the scent. It has sat on my sun porch for about 10 years serving as a place to store clothes and items we plan to donate.
But now, its (hopeful) transformation had begun. Google was a big help in both talking about what people have used, and what has not worked to get rid of the mothball scent. I emptied it out, and lightly sanded down the inside. Apparently, the camphor gets into the crevices of the wood and holds on tight. The next step was to put something called “bad air sponge” in there. I’ve read mixed reviews about this product. Mainly people complain that the scent from the sponge can be institutional and worse than the smell you are trying to eliminate. I plan on keeping this product in the closed chest for a couple of weeks, then switching it out for some activated charcoal to get rid of an new odor that might be in there. There may be more sanding in between.
I’m hoping this works because I would love to fix up the outside and get the chest back into true working order.
alookatthelittlethings 
You might also try just dragging it out into the sun on a hot day. I bought an old Singer Featherweight sewing machine and the case smelled awful–very mildewed. I put it outside, open, and let the sun bake down on it. That cleared the smell almost entirely and it has stayed gone!
I would like to try that too. I need a day when my husband is home to help drag it. I’m not sure I could navigate the steps with the size and weight.