Did ESD fry my REF200 current source circuit?
ESD may fry a component in a board, but it's less common. What happens is that the potential damage is inversely proportional to the area of the terminal you touch. When mounted, the copper tracks and vias increase this; also you have paths to ground through passive components that reduce the effective voltage produced by the ESD discharge on it.

But I don't think it could so easily happen unless you touch the terminal or a metal conductor that leads to it; air has about 1000V of maximum isolation per mm, and materials like rubber are even higher than that. You'd need a huge charge in your body.

Anyway, the usual way to protect against ESD is to place an inverted TVS diode between any vulnerable terminal and ground (by 'vulnerable' I mean those that have connections to the outside of your case).

I'd need to understand better what your friend meant when he mentioned 'grounding the system'. If it is replacing the case to a metallic one connected to ground, yes, that'd help a lot. It'd be much more probable that when you touch the device the discharge would go directly through the case to ground than through a vulnerable contact from the circuit.