It's imposible.
A picture that is 300x300 pixels means that there are only 300x300 pixels in the picture, you can't stretch it to 600x600 since the information is simply not there.
You can't just click "make into vector" becaues vectors are much more complicated than that.
If you have a normal picture from a camera or something and you would want to vectorize it... well... it can't be done, since that's not how vectors work. Vectors are shapes, rather than pixels, vectors never have any size at all they just have the information of the shape they take. So if you would want to stretch a 300x300 picture well, it still would simply not work using vectors because again, the information isn't there. And even if you try, granted it would probobly look better stretching it as a vector, it would take a grand-master proffesional vector GOD to make actual realisitc looking vectors, since it's simply something that isn't done.