Agree, as much as the OP would hate leaving, that is a possibility if the application under H&C doesn't succeed.
@ OP : Its much better to do some research, get informed and come up with a plan of what you will do if you had to return back to your country than to be forced to leave and not to have a plan.
You may be thinking that Canada is your one and only option, but the moment you shift your mindset and start looking at other potential options, you'll realize there could be an option b, c, d...etc. I understand you are a citizen of Portugal, if you don't necessarily want to live in Portugal - what about completing your studies in another EU country that may provide free tuition/or low tuition for EU citizens for example - have you thought about it?
Try to correct your status in Canada through a lawyer but in the meantime understand your other options, at the end either A or B...you'll should be able to make a plan that works for you...but continuing without status is definitely not the best choice.