Question: I thought that: "During a week" means any week and "During the week" means a specific week. 

> 
> 1) What do you eat during a week? - (*every week*)  
> 
>  2) What do you eat during the week you spend in Spain every year?
> 
> 
> 

However, it seems to me that some people use "during the week" to mean "during a week". 

Is it possible for "during the week" to mean "any week"?

Answer: 
The can also be used to represent the entire class.

> 
> The cow is a domestic animal.
>  A cow is a domestic animal.
> 
> 
> 

In the first sentence **the** represents the entire class whereas in the second one **a** is used to mean any.

 Therefore the sentence

> 
> What do you eat during the week? 
> 
> 
> 

Can also mean during every week
Yes, what you mentioned that you think is correct. "**The**" article is used to point out a specific week while **"A"** is used for any week. [(Source)](http://www.butte.edu/departments/cas/tipsheets/grammar/articles.html).  

"During the week" can not mean "during any week".