The process involves (if you get it right) the rusty object being connected to the negative supply – this is the cathode. What happens is that there is hydrogen evolved on its surface, and a smaller amount of oxygen at the other electrode (the anode). Along with the hydrogen is a process of reduction (turning ferric compounds into ferrous, for example) – this turns the red rust black, and it is loosened.(Fe2O3 becomes Fe3O4). I wish this system could do subscript-letters! This is the same black powder you get in the bottom of your central heating radiators, incidentally.
At the anode, there tends to be an oxidation reaction, so in ideal conditions metal might be dissolved. But the solution (the electrolyte) is sodium carbonate, (so if any metal is attacked it might become a carbonate) – except that the process is very fussy – difficult to get to work well – even with more suitable compounds rather than washing soda. That is why nickel and chrome plating are so expensive. Most metals and most solutions (including Na2CO3 – sodium carbonate) just won't play the game.
So, an anode which cannot be attacked is ideal (but not vital) – and carbon would meet this need, if you can find it in big enough sheets. Also good are stainless steel and lead sheet, and lots of other metals. You are, after all, concerned with the object, and you will rinse it off before working on it. You are not going to immerse yourself in the liquid or drink it, so there is little chance of poisoning yourself.
Hope this helps – Tim