Aarhusbolig
This website is a cooperation between all social housing associations in Aarhus, who together have a large number of apartments (approx. 45,000 in total) at their disposal.

You need to sign-up, pay a fee of 100kr every year and select the apartments that you would like to apply for. You will then be placed on a waiting list for the residences you have selected.

You can also find some apartments that allow you to get further ahead in the queue if you can fulfill certain criteria, such as being employed, applicants above 55 without children living at home, etc. On the page you can see the different criteria for the different apartments.  

Pros
Housing via the social housing associations are some of the cheapest, if not the cheapest, types of housing you can get in Aarhus. This also makes it extremely sought after and leads to long waiting lists as explained below. Housing in the outskirts of Aarhus might be available also after few months at the waiting list. The waiting list is administered uncorruptly and you don't risk scams.

Cons
Unfortunately, the waiting lists can be extremely long (very often from 3-20 years), so this is not a good option if you are staying in Aarhus for a short while only. If you plan to stay for a long duration, you could sign up for a membership. The homepage is in Danish only, and you need to use Google translate to understand it if you don't speak Danish. Only unfurnished accommodation, and sometimes fridge, stove etc. are not provided.

The Helsingforsgade 5 apartments are prioritized for international PhD students and Postdocs (solo or with families). The estate has a very convenient location near Aarhus University, the city centre, and shopping opportunities.

Please note that there can be a long waiting list, as only a limited amount of apartments are available.

Find more information about the place and the application procedure here.