To begin with, WE have shallow depth-of field imagers, called EYES. Our foveal (sharp image) area is only within about 4 degrees of the center of the visual field. It's up to our brains, and the rapidly-scanning saccadic movements of the eyes to put together the visual experience of a virtually unlimited DoF. In this regard, you might say our "stills" are shallow, but our "movies" are deep.
Of course, room-sized camera obscura, the grandaddies of all photography, rendered extremely deep-focus images, because image field was so huge compared to aperture. The problem with the camera obscura was that it had no way to capture the image (...this reminds us too that the camera obscura (hidden room) was a movie).
Once we learned how to capture a slice of visual time and feeze it chemically, we inherited shallow DoF again along with the technology. Before film, images were captured on large format plates. These early emulsions were relatively slow, so camera designs required either unbearably long exposure times or huge aperatures and huge lenses that could gather a lot of light fast. Perhaps the the most famous Daguerrotype of all showed about 1/4 inch of Abe Lincoln's nose in tight focus, with his ears way out there in blurryville. As film arrived in large medium formats, lens and emulsion technology made deeper DoF possible, when we wanted it.
The irony is that the "arty" look once imposed by the limitations of technology is hard to come by with today's super-deep focus digital cameras. These too, carry limitations imposed by the technology: the small size of the imaging area relative to aperture. Now folk seem to be adding the "old" look in post, with Photoshop.
One cute new DoF device for the digital still camera is the lensbabies device
On the motion side, we are still a long way from the shallow DoF or "35 mm" look, mainly because of the imaging area available on a 1/3 inch chip. Even the top-end cameras like the Cine Alta and the Viper have only 2/3" imaging areas. HD movies, such as Collateral (2004) made the most of the deep focus and high light sensitivity capabilities of the digital camera.
But artists will still prefer the film-like, "soft" image aesthetic for many projects. We are already seeing a lot of DoF play in post-work, in movies as well as stills alluded to above. The upcoming Red camera's mysterium chip will boast an imaging field larger than that of a 35 mm film camera, so we may see the blurry background look arrive in digital movies at last.