Many years ago there used to be a shop/office in the calle Elvira called  mitzfahrzentralen.

It was an agency that could arrange car shares.

For example if you were going to go to Madrid the following week you could find people to come with you and share the petrol cost.

It was a sort of controlled hitchhiking.

Does this sort of system still exist?

Do you know any web sites which do it?

I have done some searching on internet and have come up with the following.

The only one I know personally is bla bla car and it works very well. The only problem with  bla bla car is when people let you down at the last moment so it is a good idea to have a plan B. 

blablacar
https://blablacar.com/
I have known several people who have used this and they have had good experiences. This is the biggest and most well know site.