For joinery on such a small box, I think you will want to go for box joints. They will be easier to work with and you won't have to mess around creating small dovetails in thin wood where tear out will be a concern. As for saws, a small bandsaw could do well here, but making box joints on such a saw can be a bit of a problem as bandsaw blades can shift when making cuts. Cheap, small bandsaws don't usually have great blade guides and the thin blades flex a lot which could lead to accuracy problems cutting miniature box joints. I'd just rough out the joints and leave extra wood to fix with files so you can fine tune the fingers and fit rather than just scarfing out the final joint and discovering you have bad fit and gaps afterwards. It's good to have saw options if you do somewhat regular woodworking projects, but only spend the money for a new saw if you think you can get a return on it with other projects too. A laser cutter would be great here for making small boxes, but since this is a jewelry box and you probably want to use a nice hardwood, a laser cutter won't cut the denser, thicker wood. But if you want it made of 1/8" or 1/4" Baltic birch plywood, the laser cutter is your best bet.