Learn Turkish: Çok masraf isteyen eski bir daire kiralamadık. - We didn't rent an old apartment that requires a lot of expense.

It is made from the verb istemek (to want/require) plus the subject relative suffix -(y)en. This suffix turns the verb into an adjective meaning 'who wants' or 'that requires'. Here, it describes the apartment as the thing doing the 'wanting' or 'requiring'.

I thought istemek meant 'to want'. Why is it translated as 'requires' here?

While istemek usually means 'to want' when referring to people, it means 'to require' or 'to need' when used with inanimate objects or situations. So, masraf isteyen literally means 'wanting expense', which translates naturally to 'requiring expense'.

Why is the word order Çok masraf isteyen eski bir daire instead of putting the relative clause after the apartment?

In Turkish, all adjectives must come before the noun they describe. The entire phrase Çok masraf isteyen (that requires a lot of expense) acts as one big adjective modifying the apartment. Just like the simple adjective eski (old), it must be placed before daire.

Since the apartment is the object of the verb kiralamadık, why doesn't daire have the accusative case ending?

The accusative case is only used for specific, definite objects (like 'the apartment' or 'that apartment'). Because the sentence is talking about 'an' old apartment in general (bir daire), it is an indefinite object and stays in its plain dictionary form.