Learn Turkish: Göçebeler gibi yaşamak için birkaç haftalığına karavan kiraladılar mı? - Did they rent a caravan for a few weeks to live like nomads?

It breaks down as hafta (week) + -lık (duration suffix) + -ı (3rd person possessive) + -n- (buffer) + -a (dative). The k in -lık softens to ğ because it is followed by a vowel. This specific chain of suffixes (-lığına / -liğine) is the standard formula to express that you are doing something for an intended, temporary period of time.

Could I just say birkaç hafta karavan kiraladılar mı instead of using haftalığına?

Yes, but there is a subtle difference in meaning. Birkaç hafta kiraladılar simply states the duration of the action: the rental lasted a few weeks. Using birkaç haftalığına emphasizes their intent at the moment they rented it: they rented the caravan with the plan of keeping it for a temporary span of a few weeks.

Since the English translation is "a few weeks", why isn't there a plural suffix on hafta?

In Turkish, nouns that come right after numbers or quantifiers like birkaç (a few), çok (many), or az (little/few) must remain in the singular form. Therefore, it is always birkaç hafta, never birkaç haftalar.

Why is the direct object karavan not marked with the accusative suffix to make it karavanı?

Because they rented "a caravan" (general and indefinite), not "the caravan" (specific and definite). Only specific direct objects get the accusative case suffix. When the object is indefinite, it stays in its bare dictionary form and sits right next to the verb.