Oh, I remember those times well. Just before that era I was hobby-programming microprocessors using dip switches and single multi-segment LED for the output. I completely understand how Bill Gates felt when he said that and I agreed with him. Turns out we were both wrong in an epic manner.
I can't say what the bandwidth and memory requirements of software will be like twenty years from now. But I am imagining uploaded consciousness (at least lobsters, lol) or complete telepresence and those will take stunning amounts of bits or qubits or whatever form the data will take. I often wonder if we'll go back to analog.
Dennis, isn't popping the hood one of the basic requirements of hard SF?
Steve, that is an awkward bridge to cross. The best answers do seem to avoid specifying a value but then the reader has a hard time envisioning the amount of data you want to impress them with. Perhaps describing the technology (e.g. ferromagnetic, optical disk, nanoscale cross bars, optical cube, atomic entanglement) rather the meaurement of bits is better indicator of just how advanced your hardware is?