Can my father sponsor me for Schengen visa?

How will you prove you will come back?

That is the question you have to answer, to the satisfaction of the immigration officer during your interview and when filling your application.

As a fresh graduate with no job, you have no real reason to come back. You could happily stay in Spain as long as your father is willing to support you. However, this is not allowed - and will most likely get your application rejected, since its de-facto immigration.

Remember, the officer has to assume everyone is trying to game the system to immigrate to Spain; and it is your responsibility to prove otherwise - and it takes more than a smile to do that.

You have to provide evidence of home ties. These include a steady job, property, family (ie, you are married); business that you run, etc.

If you are traveling for tourism/leisure you also need to provide an itinerary, hotel bookings, flight tickets, possibly even other information (depending on where you are applying from and the strength of your documents/application).

If you are traveling for business, then you need to provide other supporting documents - like a letter from the party inviting you, proof that you and they have a legitimate business relationship, proof that you have sufficient funds to cover the trip, etc. etc.

Saying that you are going there just to stay for a while, maybe tour the place, and then possibly join a startup is a big giant NO - unless you are invited by a startup who is sponsoring your work visa in Spain.

To answer your question directly, if your father intends to support your visit financially, he needs to provide proof of sufficient funds - but as you are the one traveling and not your father, the application will be decided based on the likelihood that YOU will overstay or otherwise violate the terms of the visa you are applying for.