First, as others have noted, the human visual system has a field of view approaching 180°, but acuity is limited to a very small central region. There's actually an excellent XKCD
devoted to the subject.
The photographic definition of a "normal" focal length being equal to the diagonal of the sensor / film format has much more to do with perspective than anything else. It just so happens that a comfortable viewing distance varies with image size, such that you tend to preserve the same angle of view regardless of print size. That is, you'll stick your nose in a 4" x 6" print, you'll look at an 8" x 10" print from normal reading distance, you'll stand back to look at a 24" x 36" print, and you'll look at that billboard from a block away.
And, at that typical viewing distance, the perspective of a normal focal length lens is such that it pretty much matches the same perspective the photographer saw. That is, if you're using a normal lens, if you make an instant 8" x 10" print right there and hold it in front of you at reading distance, the print is going to match up very closely with the original scene.
You know how wide-angle lenses tend to give a distorted perspective? Well, stick your nose in the print, and it suddenly doesn't look so distorted. Same thing with telephoto lenses and their compression; stand back a ways, and it looks quite normal.
So...short version, is, there is no camera that duplicates human vision. Cameras and eyes are two radically different imaging systems. Each can do things the other can't.
however, encourage you to spend a lot of time experimenting with the matter. Pick some sort of still life and get really friendly with it. Shoot it from all different positions with all different lenses -- anything and everything you can get your hands on, even if only by borrowing and / or renting. Shoot at different apertures, while you're at it. And, most importantly -- though you might not realize it right now -- play with the light. Just a couple cheap task or work lights, a bulb with a reflector, will do -- and extra bonus points if you can dim them. Move the lights all around the scene as you move around the scene.
The purpose of this exercise, of course, isn't to create great art. Rather, it's to get a visceral, hands-on understanding of what the relationship is between perspective and position and focal length and how light plays together and all that stuff. Don't worry...once you see some of these things, you'll have more than one "ah-HA!" moment, and that'll inspire you to go do something truly creative.
P.S. Your homework assignment: find out the actual dimensions of your camera's sensor, including the diagonal. It's in the manual and on the manufacturer's Web site, along with other places. Then, cut a hole in a piece of paper the same size as said sensor. Now, hold that paper up to your eye. Have a ruler handy so you can see how far the cutout is from your eye, and compare with the numbers on your camera's lens. b&