Learn Russian: Я хочу достать мяч из-под самой большой машины. - I want to get the ball from under the biggest car.

из-под is a compound preposition meaning "from under" and it always requires the Genitive case. That is why the phrase following it, самая большая машина (the biggest car), changes to its Genitive form: самой большой машины.

Why use самой большой to mean "the biggest"?

To form a superlative (the "most" or "-est" version of an adjective) in Russian, you often use the word самый (the most) plus the regular adjective. Here, both words must be in the Genitive feminine form to agree with машины, resulting in самой большой.

Could I just use взять (to take) instead of достать?

You could, but достать is much better here. It specifically implies getting, extracting, or reaching for something that takes physical effort to access — exactly what you would do to rescue a ball stuck under a car.

мяч is the direct object being retrieved. Why doesn't its ending change to show the Accusative case?

It actually is in the Accusative case! Because мяч is an inanimate masculine noun, its Accusative form looks exactly the same as its Nominative (dictionary) form.