Learn Russian: Их переговорная занята, мы сами зайдём в его кабинет. - Their meeting room is occupied, we'll drop by his office ourselves.

Unlike words for 'my' or 'your', the third-person possessive pronouns его (his/its), её (her), and их (their) are completely unchangeable. They are actually the genitive forms of the personal pronouns 'he', 'she', and 'they'. So, even though переговорная is feminine and в кабинет is in the accusative case, их and его stay exactly the same.

переговорная looks like an adjective ending in -ая. Why is it used as a noun here?

Many Russian words for specific rooms are actually adjectives that act as nouns, with the word комната (room) implied but dropped over time. переговорная комната simply became переговорная (meeting room). You still decline it exactly like a regular feminine adjective.

Why is the short form занята used here instead of the full adjective занятая?

занята is the short form of the adjective занятый (occupied/busy). In Russian, when an adjective is the main state of the sentence (the predicate), especially for a temporary state like a room being currently in use, the short form is preferred. It takes the feminine ending to match переговорная.

Why use сами for 'ourselves' instead of a form of the reflexive pronoun себя?