Learn Croatian: Zgrada u kojoj živimo ima velik zajednički prostor. - The building we live in has a large shared space.

The relative pronoun koji has to agree with the noun it refers to, which is zgrada (a feminine noun). Because it follows the preposition u (in) indicating a location, it needs to be in the locative case. The feminine singular locative form of koji is kojoj.

In English we say "the building we live in". Why can't we say Zgrada koju živimo u?

In English, it's common to leave a preposition at the end of a relative clause and drop the word "that" or "which". In Croatian, you can never leave a preposition dangling at the end, and you cannot skip the relative pronoun. You must always structure it as "the building in which we live" (Zgrada u kojoj živimo).

Why doesn't the direct object velik zajednički prostor have a special accusative ending like -og?

The phrase is the direct object of the verb ima (has), so it is indeed in the accusative case. However, prostor is a masculine inanimate noun (a thing, not a living being). For masculine inanimate nouns, the accusative case looks exactly the same as the nominative case, and its adjectives (velik, zajednički) follow that same rule.

Why is the adjective velik rather than veliki?

Velik is the indefinite form of the adjective, which we use here because the sentence translates to "a large shared space" rather than a specific one. If you used veliki (the definite form), it would imply the building has "the" large shared space.