All over Europe you will find small B&Bs (not the Laura Ashley style, but often just 2-4 rooms) or private rooms or a small apt rented out by private persons in their house. Normally breakfast is included and very often you could use a kitchen for some cooking if you want.
Search the websites of the city's tourist information for contact data and pricing information. If you rent for more than some days, pricing is often up to negotiations.
In some countries youth hostels do not have age restrictions and offer a private room with own bathroom for a couple or family.
Or try subletting, for example via http://www.mitwohnzentrale.de/staedteliste.php
Many students try to sublet their appartment while they spend a semester abraod or have an internship in another city.
You will also find regular holiday homes all over Europe.
Try to avoid July and August due to school holidays.