What length can a metal wire (cable) hang vertically? What length can a cable hang vertically with no additional weight attached to it. 
I tried figuring this out with 5/8 aircraft cable specs by dividing the rated weight capacity of the cable by the actual weight of the cable per foot. This lets me know how many feet of cable it takes to equal the cable's rated max capacity. 
But that doesn't seem right. Can anyone help?
 A: Perhaps you are overthinking it a bit.  You said in the comments:

I don't know much about the physics of cable, but I'm sure the weight rating for a load on the cable and the cable hanging free are different.

The weight rating is just a measure of the maximum load the cable can handle (presumably in tension, since that is generally what cables are used for).  It's based on the maximum stress the material can handle before failure (probably with a healthy safety factor added on).  It can be measured in terms of force instead of stress, because the cross sectional area of the cable should remain constant throughout the length; and therefore any force acting on the cable will be directly proportional to the stress.
This means that at any point in the cable, it is rated to be able to withstand that force acting on it.  Due to the nature of a self-weighting cable; it will have a minimum force at the bottom of the cable.  The bottom of the cable isn't being stretched, because there is no weight below it pulling it down.  The middle of the cable will have a tension force equal to half the weight of the cable, because it has to support the weight of the half of the cable below it.
The very top of the cable, where it is connecting to whatever is holding it in place, has a force acting on it equal to the entire weight of the cable.  This means that if the total length of cable weighs more than the rated strength of the cable, it may fail at the very top of the cable.
Basically, having more cable below adds the same forces to the cable that having an equivalent weight would add, so that calculation you described makes sense to me.
A: If I search for 5/8" aircraft wire I can find a product weighing 0.65lb per foot with a strength of 37,000lbs. That gives a length of about 10 miles using the calculation that I assume you used.
The theoretical length of cable that can support its own weight is related to the Specific Strength of the material (aka the strength-to-weight ratio). The linked Wikipedia article gives a table of values for various materials, which I imagine are theoretical limits for a perfect cable.
Although not specifically given, the article implies that piano wire could support tens of kilometers of its own length. Other steels (presumably not designed for tensile strength) have breaking lengths of only a few kilometers (e.g., stainless steel, 6km).
But nevertheless, these figures are consistent with the result above.
