Here’s how I make an oval box. This one is made from 20# EZboard 1.5″ thick (HDU) ezcarving.com. 20# density, 1/2″ thick top and bottom. You can do the same with wood but you’d need to use a narrow band saw blade rather than a scroll saw.
1. cut the outside oval on scroll saw
2. cut the inside oval
3. glue the oval to an oval for the bottom. I use 
Go2Glue and it works really well on HDU.
4. Resaw thin piece from center cut out for underneath the lid. Use cut offs to stabilize piece while resawing
5. glue this piece underneath an oval for the lid
6. Fit the lid into the base. Some adjustments may be needed to form a good fit
7. With the top on the box, sand the outside smooth on a belt sander.
It’s quite fast to make an oval box like this from HDU because the material is all uniform thickness. This box will be chip carved 