I don't know. How long are weeks doesn't really sound natural to me. "How long is the week" or "How are long are the weeks" sounds slightly more natural.
Would "How long do the weeks take?" also be acceptable in this case or does that have a different translation altogether? This sentence has just really baffled my brain. It probably doesn't help my brain is feeling quite mushy right now.
You could say either "How long do weeks last?" or "How long does a week last?" Personally, I feel that "How long are weeks?" and "How long is a week?" sound a little more natural, but only slightly so.
Is that exact expression common, though? Well, I guess most people nowadays are only concerned with 7-day weeks, even though many other week lengths have been used throughout history, so I would have to say no – it's not common. =)