According to the DLE
entre pecho y espalda
- loc. adv. coloq. En el estómago.
In English
between chest and back
- Into the stomach
As you pointed, usually this expression is used with food to remark the consumption of a great quantity ("stuffing oneself").
If you check the meaning of binge-watching in the Wikipedia in Spanish, you'll find the term atracón
Un maratón de series, atracón de series, y las expresiones inglesas binge-watching, binge-viewing o marathon-viewing, son términos que hacen referencia a la acción de ver varios capítulos de la misma serie de televisión de forma continua
In this term we can find the key of your question
m. Exceso en una actividad cualquiera. Atracón de trabajar, de llorar.
m. coloq. Acción y efecto de atracar (‖ comer y beber con exceso).
Being the main use of atracón an excess on eating and drinking (2), the term can be used with an excess on any activity (1).
Summarizing
Nos hemos metido entre el pecho y la espalda unas veintialgo películas.
=
Nos hemos dado un atracón de unas veintialgo películas
= EN
We have watched twentysomething movies in a row
=
We have binge-watched twentysomething movies
Last but not least, the expression gathered in the DLE does not show the articles present in your sentence, el and la, as Charlie has pointed in the comments. I agree with him, the most common form is entre pecho y espalda, with no articles.