According to the link, the drawers are 11" deep. That's way too deep for cones of thread IMO. I currently use several plastic storage drawer sets, similar to these. I got mine before I started accumulating thread cones, so I have other things stored in the shorter drawers. Just measure the cones to find out what height drawers you need.
https://www.walmart.com/ip/Sterilite...Tower/15442448