Such a thing is seemingly impossible with a traditional camera.

I disagree with the premise of your question. People take well-exposed photos that include the sun with DSLR's all the time. If you're using a very wide angle lens outdoors, you may not be able to avoid having the sun in the frame. And yet, there are still plenty of blue sky images. Here's just one example from Flickr.

Background

The sun can cause exposure problems for you if you're shooting in an automatic exposure mode. For example, if you're shooting in aperture priority mode, having a very bright object like the sun in the frame may cause the camera to try to balance the exposure by using a faster shutter speed than you'd need to expose the rest of the frame correctly, so you get an underexposed image. In shutter priority, the camera may choose a smaller aperture for the same reason. There are lots of ways to deal with the problem, including:

• shoot in manual exposure mode: If you're controlling shutter speed, aperture, and ISO, then the exposure will be whatever you decide. If your first shot is too dark, adjust one or more of those three parameters until the part of the frame that you're interested in is properly exposed.

• exposure compensation: Any DSLR will have some means for you to tell the camera's auto exposure system that you it to adjust the exposure so that it's darker or lighter than what it would choose on its own. That lets you still use Av or Tv modes while getting the exposure you want.

• metering modes: As discussed above, the reason that your shots are improperly exposed is that the metering system is giving too much weight to the very bright sun. If you switch to a center-weighted or spot metering system (there will be a setting on your camera for the metering mode), you can effectively tell the camera to expose a specific part of the frame correctly. Read your manual for camera-specific instructions, but in these modes the camera will use either the center of the shot or the area around the current AF point.

• auto exposure lock: This is a camera feature that lets you lock in a specific exposure and then reframe before taking the shot. On a Canon, for example, you could set the camera to spot metering mode, point the camera at the part of the frame that you want exposed correctly, press the AE lock button, then reframe and take the shot. This even works with evaluative metering -- you can point the camera in a direction that doesn't include the sun, hit AE lock, reframe and shoot. Think of it as a sort of AE override.

• wider field of view: If you use a wide angle lens (or the wide end of a zoom, of course), you make the sun a much smaller part of the image, which means that its effect on the metering system is diminished.

• filters: Adding a filter to your lens lets you modify the image before it gets to the sensor. If you want to darken the sky, two types of filters that will help are:

  • graduated neutral density: A GND filter is darker at one edge than at the other. If you position it so that the dark part is at the top, you can block some of the light from the sun and sky, preventing the sky from being more exposed than you want.
  • polarizing filter: Light from the sky is partially polarized, so you can use a polarizing filter to darken the sky as well as to reduce glare in other parts of the image. A polarizer won't help with the sun specifically, but it will prevent an overexposed sky on a hazy day. For a DSLR, you'll normally want a "circular polarizing" (CPL) filter.

Answer

The reason that the GoPro takes nice shots that include the sun isn't that the sensor has greater dynamic range than a DSLR, it's that the camera is tuned to give good results with minimal fuss. It probably makes some assumptions about how to react when faced with very bright objects that are different from what makes sense for a DSLR. And the field of view ranges between 170° and 85° depending on setting, which is equivalent to a 12mm-24mm lens in 35mm full frame terms. With such a wide field, the sun isn't likely to blow out the image.

From what I've read, a GoPro Hero 3 has a dynamic range of about 11 stops, which is comparable to a typical DSLR. If it had a lot more range, then you could take shots where very bright and very dark objects both have visible detail -- that's a hallmark of HDR photography (where you combine several exposures to get detail across a wider range), and it's not what you see in GoPro shots.

TL/DR

It's easy to take shots that include the sun on a GoPro because of the wide lens and different metering system, but you can take shots that are just as good or better with your DSLR once you know how to control it.