Aarhus University has a guesthouse for visitors needing accommodation for a few days or weeks:
The Guest Department at the Mathematical Department
The guest department at the mathematical department offers six guest apartments and twelve guest rooms on campus.
Visit Aarhus is the official tourist office of Aarhus and has compiled an extensive overview of hotels in Aarhus.
Visit Aarhus also has an extensive compilation of Bed & Breakfasts in the Aarhus area.
Bed & Breakfast in Denmark also shows a number of additional B&B's - marked on a map with contact details and prices.
Via Airbnb you can find plenty of people who typically rent out their spare room or entire apartment for short durations of time. It's can be a cheap and more personal alternative to hotels & traditional B&B's.
Other B&Bs not featured on Visit Aarhus' site:
If you are low on money, urgently need a place to stay for just a few days and are feeling very adventurous then you can consider "Couchsurfing".
Via the Couchsurfing community it is possible to find local hosts who are willing to offer a place to sleep for free. You can set up a free account and get in contact with hosts in Aarhus.