¿How long are you going to live in Madrid?
If you are staying for many months or years, maybe you shoul rent a flat upon the law "Ley de Arrendamientos de España", but in that case you must remain in the flat during the entire rental period, one year minimum.
Another option is to search accommodation in a hotel or hostal, or to rent for days a self catering apartment.
A self catering apartment have many other advantages and aditional services: internet, linen, cleaning...
If you finally decide for a short term rental in Madrid, here you have some recommendatios: Self catering Madrid in Conde Duque area or this Self catering Madrid apartment in the center of Madrid. Red more about that kind of accommodation
Buying a house in Madrid is an expensive option.
If you prefer a 5 stars hotel or a hostel, it will have the disadvantage that this kind of accommodation has no kitchen, no lounge, less intimacy or privacity.
Another option for the students is to find a roommate.