Here's some napkin math on the black hole argument. Information is essentially equivalent to entropy: the entropy S = k * log M, where k is Boltzmann's constant and log M is the number of bits encoding the information. Now, it turns out to be fundamentally impossible to store more entropy in any region of space than would be contained in a black hole of the same size: S_max = (k A c^3)/(4 G hbar), where A is the "surface area" of the region, c is the speed of light, G is Newton's gravitational constant, and hbar is Planck's constant. That means the maximum number of bits of information in a region of space is (A c^3)/(4 G hbar), or about A * (10^69 1/m^2). Our solar system is maybe 1 light year in size: about 10^16 m, so its "surface area" is 4pi r^2 ~ 10^33 m^2. That means that the solar system can hold at most 10^102 bits of information.

If Moore's law says that computer memory doubles in capacity every 2 years, then they'll go up by a factor of 1000 about every 20 years. We currently have computers that hold about 10^12 bits of information, so just 90*20 = 1800 years would get up to that absolute limit, regardless of what technology we used, unless our single computer was bigger than a light year in radius.