Aside from being a very popular dance, "chacha" is a shortened version of "la muchacha".
So if we translate our sentence we come up with something like:
What? You want me to be the girl?
Well, eso no tiene sentido - that doesn't make any sense. But it will after I give you this key piece of information.
"La muchacha" is way of referring to what we might call a nanny - someone who helps out with the kids and chores around the house (las tareas de la casa). So let's take a look at our sentence again.
¿Qué? ¿Quieres que sea tu chacha?
What? You want me to be your maid?
Although chacha, or la muchacha, is more accurately defined as nanny, in this context it's best translated as maid. This is something you might hear from a wife or girlfriend who is unhappy about having to constantly clean up after their significant other all the time, which is how I first heard it. I always get a good laugh when I hear the word chacha, because it reminds of my amiga ranting about her novio (boyfriend), who she affectionately calls "el gringo" or "cabrón".
Other terms you can use are empleada doméstica, sirvienta, mucama, or afanadora. I don't really recommend using sirvienta, because Spanish speakers seem to universally agree that it's a bit disrespectful.
Well, that's it for today. Hasta la próxima.