If you are looking for a self catering holiday in Madrid, the best option will probably be to look for a holiday apartment, at least if you want to stay near the city center.
Another possibility would be to look for a holiday villa in the outskirts of Madrid or in some of the surrounding towns, where the choices are plentiful. Holiday villas are available with many different facilities, like swimming pool and air conditioning in different sizes and price levels to match most budgets.
You can book your holiday villa through an agency or direct with the owners. There are many search engines for both alternatives.
If you book through an agency, it is usually more expensive, but the agency will normally guaranty the standard and the equipment of the villa. If you book direct with the owner you only has his word for the standard of the villa.