The Schwarzschild radius of a mass is the size that the mass must be compressed to for it to become a black hole. For our sun, the Schwarzschild radius is about 3 km, much smaller than the sun's radius, so the sun is not a black hole.
Black holes are formed when stars much larger than our sun collapse under their own gravity until they are compressed to a size smaller than their Schwarzschild radius. At that point, the event horizon of the black hole forms at the radius. Our own universe appears to contain enough matter so that it is effectively in the interior of a black hole, with the event horizon occurring at the universe's cosmological horizon.
There is an upper limit to the amount of information contained in a black hole, and it is determined not by the volume of the black hole, but by the area of the two-dimensional surface of its event horizon. For our universe, this upper limit is estimated to be 10^122 bits. An incomprehensibly powerful computer (located outside of our universe) could store the state of the universe at a particular instant by recording the state of these 10^122 bits.
Time is not infinitely divisible. The smallest unit of time that has any meaning is Planck time, the time it takes light to travel a distance of one Planck length. Storing the state of the universe at each interval of Plank time would enable the simulation of the universe, by playing back the states in order. Constructing such a computer would be tantamount to constructing the universe.
It is not possible to build such a computer because the state of the universe cannot be completely determined, due to the Heisenberg Uncertainty Principal. There is a way, however, to build a computer to accomplish the same thing without having to determine or store any configurations of the universe.
All that is necessary is to build a computer that counts in binary. Given an infinite amount of time, the computer will eventually count up to a configuration state of our universe. In fact, the computer will eventually count to a vast number that represents all successive configurations of our universe, from the first instant of creation to the last evaporating black hole. In the process of counting to infinity, it will not only calculate our entire universe's full history, but all possible universes and their histories.