with a soft metal like copper the edges could be routed so the transition of the metal blends with the substrate. For this to work well a decent substrate would be needed ...like butcher block and the copper would need to be firmly bound to the wood.
Typically any metal counter has soldered or welded corners that are ground smooth and sanded/polished to make the corner appear to be seamless. Copper is the easiest to do. However copper sheets are expensive.
This is not really hard to do, however for a DIY person that has not done this work before, it can become a bit difficult to achieve a professional look. Even those who work with these metals have difficult times unless outfitted with proper bending and cutting tools. Not your normal every day over the counter tools.