Large, probably 10,000 + tons.

 

The ship will be engaging an activity that will make its location fairly obvious and probably require it to be closer to possible attackers than the group would be without it, which will cause group commanders to want it deployed away from the rest of the group when firing.  So it will need to mount a layered defense against aircraft, cruise missile, submarine, and small boat attacks.  It needs to be able to supply enough power for the guns.  And it be massive enough to handle the recoil from the rail guns fired at 90 degrees off the bow.

10k tons is not that large...it's destroyer sized.  So rainguns railguns will be on ships roughly the same size as our current surface combatants?

Your right, it is not large enough.  Fortunately I said 10,000+, and it will be a large '+'.

 

After reviewing the data let's make it 14,000+ tons.  Figure 10,000 tons for the basic ship with the defensive systems needed to survive on its own.  Then each of the rail guns will add about 2000 tons, with over half of the total weight being in the magazines for a lot of ammo and automatic handling systems.