There are a couple things you can do, depending on what kind of look you are willing to accept.
- wood filler and then a VERY light sanding (try 800 grit and above) to try and get it flush and smooth
- if you have scrap cherry plywood you can do a little sanding with that to get the sawdust, mix it with woodglue and smooth it on top of the dents and then sand flush and smooth
- you can sand carefully and TRY and stop before you go all the way through the veneer (that veneer is very thin so it will be tough to do)
The wood filler option will most likely look COMPLETELY different from the rest of the box unless you are super careful about color selection, and REALLY lucky on placement. The sawdust/glue option could be passable, but still will look “off” from the rest. The sanding on top of the veneer can potentially look either the best (if you don’t go all the way through the veneer) or the worst (if you do go through the veneer).
I would then refinish the area (or the whole box) with your finish of choice. I’ve used Lacquer on mine, but you can do a number of different finishes for this kind of project.