Hello Blomme and welcome to our ALL SPAIN Message Board!
Most all hostels (called "hostales" in Spanish) are family friendly but the truth is few hostel rooms are big enough for 4-persons. However, since your children as so small it might not be so difficult. Will you have a kind of baby bed for the baby? A number of hostels have rooms with 3-beds.
Hostal López comes to mind. It's very very close to the Prado Museum AND Retiro park and is located on a pedestrian street in the very lively Huertas neighborhood which has lots of nice restaurants and tapas bars. They have one large-ish room with 3-beds, a balcony, and a private bathroom. The room is simple but the location and the family who own/operate are very very nice. The daughter, Nuria, speaks English too.
Saludos, MadridMan
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