I'm with pim. I've never heard of it either outside of those for students and even those are hard to find if you're not associated with some school/academy.
As pim suggested, you might try an Aparthotel (these can be expensive though, per night) and many have been listed here in this forum (use the SEARCH link to find them).
from those, you could try the "apartment" of Hostal Dulcinea
in the Huertas district. I recall JoeSambuca stayed there a number of days while he and friends were in Madrid around Christmas and loved it. He wrote up his recommendation in this forum too.
There's also the possibility of the somewhat pricey B & Bs of Madrid (also almost an unheard of concept in Spain) @ www.MadridBandB.com
. I don't know anyone who has stayed at one though. Good luck!
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