You need to go careful with this because you can really mess things up in the final straight.
An oil finish sounds just the job, except the smell will linger for a very very long time in a closed box and can be pretty stomach turning. If you can leave the box open for about ten days then the Osmo smell will dissipate, and that's by far the fastest I've found. Other oil or oil/varnish finishes might take many months. I almost never use them on the inside of cabinets for just this reason, although if it's a long build I might use Osmo provided cab doors and backs are left off for the duration of the build to let it fully harden off.
Shellac sanding sealer or a basic rubber of a well cut shellac is a simple option. The warm tone of the darker shellacs goes great with ABW. No smell after a few hours. The more coats you apply the glossier the result. Providing you don't need dead flat matt this may be your best option.
If you're just going to finish the outside you need to protect the internal/external edges with masking tape or the finish will bleed over and look messy, as you've got quite a few boxes to do that's a major faff.
Good luck!