Why do ionic compounds have higher melting and boiling points despite having smaller radii?
As pointed out in a comment, there is a problem with your second sentence:  


  "I thought that the closer the two ions are, the larger the repulsive force, and therefore it should rather require less energy to separate them..."  


This statement is technically correct, in that the repulsive forces between the two atoms do increase with decreasing bond length. And it is true that two atoms having a charge of the same polarity would exhibit increasing repulsion as the distance between them decreases.

However, in the case of the oppositely charged atoms found in ionic bonds, the attractive electrostatic forces also increase with decreasing bond length, which is the reason for the strong bonding and thus high melting points of ionic compounds.  

A discussion of the bond strength of ionic compounds is given in this Wikipedia article, which summarizes that the ionic bond strength can be determined as the sum of the attractive electrostatic energy and short-range repulsive energy.