Implementing discrete time systems
"Does this mean discrete time systems are clocked so that the values only at discrete time are available to the system?" 

yes, but the system can do some mathematics to interpolate values in between the discrete times of the samples.

"How are these discrete time systems implemented physically?"

well, it's called "electronics" and the discipline to learn is "electrical engineering".

the most essential part is a computer of some form.  DSP chips are little computers of a form that are optimized to perform the most common DSP algorithms.  you will likely also need something like an A/D converter and D/A converter, but without more specific information about what the discrete-time system is supposed to do, i wouldn't know that for sure.