Thanks Tom for rooting me from up under my rock.
That damage clearly looks like clear coat failure, if in fact you have clear coat on that car (it didn't become standard until the early 80's). I would need to know that first because the damage is going to be handled different on a single stage paint job as compared to clear coat/base coat. What you can do also depends on the shape of your paint to begin with. If the paint job on your car is fairly recent (say up to 6 years and well maintained), then you should be able to fix it without repainting. If it is the original paint, it is going to have to be resprayed.
Folks have chimed in with the suggestion of claying. Claying is a process that remove surface contamination from the paint. If the contamination you have is on the surface of the paint, clay will remove it. You may need to apply some pressure, unlike what you see most people do. In the videos that I make where I do some claying, you'll see that I use a lot more pressure than most because I don't want to be claying a car for the next 3 days. The 6-year old clay that you have will be worthless if it has dried up. If you can't kneed it like Clay-Dough, then toss it in the trash and buy some more. It doesn't matter who's clay you buy because all clay sold in America is made by the same company (Auto Wax), and everyone else just re-brands it. Like Tom brought up, under NO circumstances do you want to use that clay if it hits the floor. It will pick up something that will scratch your paint to hell and back if you try and use it again. Here's a short video of me using clay. There's no rocket science to it.
https://www.youtube.com/watch?v=P0_pFgsjlJk
If the contamination is BELOW the clear coat or into the paint, you're going to need some polishes and equipment to fix it. I would NEVER suggest that you think about wet sanding that damage given your level of experience in dealing with paint but that may be what it takes to get rid of that damage. A LESSER approach would be to use a compound followed by a polish to remove the damage, if it hasn't etched too deep into the surface. Compounds and polishes are like wet sanding with an abrasive liquid instead of sand paper. A MUCH safer alternative than using sand paper itself. In
these 5 videos, I explain the whole process of polishing with a machine using a compound followed by a polish. You can't use one of those cheap, Auto Zone style buffers that use bonnets, you have to buy a real machine (cost around $116 for the basic machine itself). Considering everything you may have to buy to attempt to fix this issue, you may want to take it to a professional detailer. Although buying everything you may need to fix this issue will set you up with everything you need to address your paint's imperfections, some folks will choose to skip that expense and go to a professional detailer. Personally, I would buy everything I needed to fix and maintain my paint myself.
So as you can see, this could be an easy fix or a rather complicated one. It depends on the level of damage you're dealing with. A picture cannot tell me that so I'm addressing every possible scenario. If you decide to get what you need to fix it yourself, I can definitely assist you in getting setup. I don't sell anything or work for any detailing companies myself (I'm a computer/network consultant), but I can help you find the stuff you need (some of it locally). So holla' if you need me.
The Junkman