When we consider a bond between two atoms, the type of bond gives a relative measure of the bond length.
Triple Bond Strongest, 3 electron pairs shared, shortest bond
Double Bond Middle, 2 electron pairs shared
Single Bond Weakest, 1 electron pair shared, longest bond
We need to determine the bond types for each molecule or ion:
CO, When drawing the lewis structure here, we end up with a triple bond between the carbon and the oxygen (and a -1 charge on the C and a +1 charge on the O)
CO2 This lewis structure gives one double bond between each oxygen and the carbon.
[CO3]2- This one is a little trickier, but when we draw the lewis structure for this ion, we get a double bond between one oxygen and the carbon, with single bonds between the other two oxygen atoms and the carbon. Because each of the oxygens want a double bond to the carbon, they all fight over it (giving resonance structures), meaning that in the end, you get a bond something between a double and a single bond for each Carbon-Oxygen bond. Therefore, this bond will be longer than a single bond, but shorter than a double bond:
So, this give the following increasing order of bond length:
[CO3]2-, CO2, CO