This could be done very easily with a SCR. Press button to fire the SCR, it will now stay on as long as sufficient DC is flowing through it, then when you press the other button, it opens the circuit, the SCR turns off, and stays off.
All you would need is a normally open switch to turn the SCR on, a normally closed switch to turn it off, and 2 resistors to connect to the gate of the SCR to fire it from the first switch.
The SCR you get would depend on what the load is, and how much current it draws.