These are only 180° shots, not 360°. The image is just two different 180° shots. You can technically create these images with any lens (ignoring issues regarding the earth's rotation).

In order to take such a photo with a single exposure, you would need an 8mm fisheye lens on a full-frame camera, or a 4.5mm fisheye lens on an APS-C camera. However, you can also simulate the same shots using a longer focal length and a panorama stitching program like PTGui or Hugin.

For instance, I only have a 15mm fisheye lens and a full-frame camera, but if I take 6 pictures with it at 60° increments -- aimed slightly above the horizon -- I can easily reproduce a full circular fisheye image. (I can probably get away with 5 images, actually.)

Even if you only have a 24mm rectilinear lens, you can still make such an image. The only challenge is that if your exposures are of a significant length, the Earth's rotation can make it difficult to stitch the images together. The longer the focal length, the more images you will need, and that makes the rotation problem even worse.