Can we separate positive ions from negative ions in a solution?

Question

Can we make a solution containing only positive ions by using electrodialysis but instead of using 2 kinds of membrane (one blocks the positive ions and one blocks the negative ions) we can use only one kind (the one which blocks the positive ions) so the positive ions wont reach the electrode and wont be electrolysed while the negative ions will be electrolysed leaving the positive ions in the solution

Answer 1

While it is true that solutions are not always locally electroneutral, that non-neutrality is small, and does not extend to larger scale. Thus, it does not impact chemical concentrations: you cannot separate positive and negative charges in a solution at a macroscopic level. See for example this excerpt from Ionic Transport Processes:in Electrochemistry and Membrane Science:

       

Answer 2

For water and on macroscopic level we can't, as there is non-zero water dissociation constant.

Dropping solvent dissociation from consideration, we should consider molar charge. Faraday's number is molar charge and it is ~10^5 Coulomb/mol. So, if you can store really much charge (at least some Coulombs) in your solution, you can. But can anybody? It is really hard.