Learn Russian: У нас нет места для большого шкафа. - We do not have room for a large wardrobe.

They are in the Genitive case because of the preposition для (for). The word для is always followed by the Genitive. Since шкаф (wardrobe) is a masculine noun, it adds -а to become шкафа. The adjective большой (large) changes to its masculine Genitive form, большого.

How do you pronounce the ending of большого?

The -ого ending in большого is pronounced as "-ovo". In Russian, the letter г in the Genitive adjective and pronoun endings -ого and -его is always pronounced like a "v" instead of a hard "g".

Why is места in the Genitive case?

The word места is the Genitive singular form of the neuter noun место (place/space). It takes the Genitive case because it comes after the negative word нет. In Russian, the thing that you do not have or that is missing must always be in the Genitive case.

Does место mean "room" as in a room in an apartment?

No, the word for a physical room in a house or building is комната. The word место translates to "room" in the sense of available "space" or "place". So нет места means "there is no space".