Q:

Could there be a rocky star, made from silicon, oxygen, etc?

Rocky in terms that it will have mostly silicon and oxygen composition, just like terrestrial planets. But not because it runs out of hydrogen and helium, rather it should start it's life by fusing silicon and oxygen. Could it, at least theoretically happen if a huge mass of silicon randomly clumps around somewhere? Could that huge mass start it's nuclear fusion and shine for a few days? If it is possible, then how much mass would be needed?

A:

Yes, why not? One could argue that gigantic stars on their death bed (couple of hours before supernova) have a lot in common with what you had in mind - big core of sillicon undergoing burning process, albeit burning here is achieved by gravitational pressure from the outer shell, which, at that point, is a funny onion-like structure with layers of H, He, C... on top and ...O, Si closer to the center. These elements were originally created by fusion, and boundaries between regions is where temperature/pressure allows for corresponding fusion process to occur.
If you would replace onion-shell of such a star with the same amount of pure Si, it would still apply the gravitational pressure, and there is no reason I can think of that would prevent you from getting a nice pure Si star. Which would still explode into a supernova in the matter of hours :)
Now, about the numbers, quick glance at wikipedia revealed sillicon-burning process starts at 2.5-3.7 billion Kelvin, which is temperature you can get at the core of stars 8-11 times massive than the Sun. 
So if you start pilling Si somewhere in space, you will not get meaningful amounts of fusion action at core until you get to round 8 solar mass. 
If you really try do this clumping in space somewhere, your clump would attract a whole bunch of other kinds of materials, so I doubt pure Si fusion star happened somewhere naturally.
Also, take note that such a clump of Si would get weird much before it reaches fusion threshold, because it would still be millions of degrees hot. It would shine a lot of energy even before it could technically be called a star. So it wouldn't really look that much rocky and terrestial.

