I have to admit that I find it a tad ironic that Michael started this thread since he's such a proponent of little cameras.
Any way, chip size, aperture (f / T stops), shutter speed, focal length, gain, back focus (macro lens) etc. All these affect, and thus give you control, of the depth of field (DoF).
I use the aperture; built in ND filters; -3, 0, and +3 gains; shutter speeds (up to around 1/100); distance to subject and from subject to background; the macro feature; whatever I can use to get control of the DoF. Some times one can use a scrim behind the subject to help put the background out of focus, but you need time, and of course the gear.
You
can crush the DoF with a small chip camera. You just need space to set up the shot. I shot this video with the JVC 110U camera which has 1/3" chips. To shoot the portrait shots I had to back pretty far away from the subject (~30') so I could zoom all the way in, and place the subject far away from their background (30' - 50' +). Then I used the ND filters to get me the widest aperture possible. In a couple of the normal B-roll shots I was able to back off and zoom in to get the background to fall off a little, but not often.
Suncoast Hospice
In this one you'll see an interview with the Chief of Police (it's near the end) in a huge room. To see most of the room in the background, we needed to put the camera and subject in a corner of the room. Because of the position, I had to leave the lens zoomed most of the way out for the wide shot. The camera, a Sony V1U, has 1/4" chips. Because of those facts, I couldn't put the background out of focus. Even though I was shooting wide open (maximum appeture), the whole background is rather sharp.
Disappeared
Some times you can use the macro option to do a nice rack. If you have a real lens, like on the JVC, or a full sized camera, you can focus on the farthest subject, and then refocus on the nearest subject with the macro ring. The background goes out of focus. When you want the far away subject to come in to focus, you just turn the ring until it snaps and poof! A nice steady rack focus. Again, it helps if you can zoom in a bit, if the subjects are significantly far apart, etc.
Some times I use the macro to put the background out of focus when shooting B-roll and won't even do the rack focus. I haven't tried it for an interview yet though. I'm afraid it might crush the DoF too much. Has any one else tried it?