Learn German: Das ist ein Fall für den Chef. - That is a case for the boss.

No, this is a classic false friend! In German, der Chef means the boss, manager, or supervisor. If you want to talk about a cook or head chef in a restaurant, you would use der Koch or der Küchenchef.

Why is it ein Fall but den Chef? Aren't they both objects in the sentence?

They are doing different grammatical jobs. The verb ist (is) works like an equals sign, so the noun after it stays in the nominative (subject) case, which is why we use ein Fall. However, Chef comes right after the preposition für (for). The word für always forces the noun following it into the accusative case, changing the masculine der Chef to den Chef.

Could I say Der ist ein Fall instead of Das ist ein Fall?

You need to use Das here. When saying "That is..." or "This is..." to point something out, German uses the neutral pronoun Das, regardless of the gender of the noun that follows. You don't need to match it to the masculine noun Fall.

How is Fall being used here? Does it mean someone fell down?

While der Fall can mean a physical fall, here it means a "case" or "situation". It is used exactly like in English when a detective or a manager has to handle a specific "case".