Is there any theorical reason why this can't be done?
It has been done. The Dallas/Maxim 1-wire protocol has power, clock, and data all on one wire.
How?
The line is normally at +5v (providing power, devices have a diode and cap to maintain it during signalling).
Each falling edge is the "clock" signal.
Each rising edge is the "data" signal - short duration low pulse = binary 1, long duration = binary 0.