How much power can reasonably be collected from ambient background radiation?
The answer is "very little".  I have also seen 10s of volts on a 1 M&Omega; scope probe when I touched it.  However, the scope can only see that voltage because it has a explicit ground reference.  The basic problem is that it will be very hard to get a large enough differential power pickup.

All voltages are effectively differential.  It worked for the scope only because it was implicitly using the ground plug as the other connection it was comparing the probe tip voltage to.  Getting a meaningful differential voltage from a small box is a much harder problem.  To get any differential voltage at all by capacitive coupling, you need a E field gradient.  That absolute value of the E field does not matter.

My knee jerk reaction therefore is that no, you won't be able to harvest enough power in a small box from stray power line fields in your house to run a microcontroller.

Anecdote:  Many years ago when I was in grade school, I had a crystal radio.  That is a type of radio that runs completely from the received RF power.  We lived about 20 miles from a 50 kW AM radio station, and I had about a 50 foot wire antenna strung accross the back yard.  With that antenna I could pick up that radio station at quite reasonable volume with headphones.  I could also have the radio drive a impedance matching transformer connected to a small speaker.  You could hear the radio station reasonably well if you put your ear up to the speaker.  It was also great fun to annoy my brother with at night.  I could leave it on, and you could sortof hear someone talking accross the room, but it wasn't loud enough to tell what they were saying and the sound was difficult to localize.  Considering how relatively loud the sound was into the speaker or the headphones, there just might be enough power there to run a micro.  The trick will be to convert it to a useful DC voltage without too much loss.  By the way, my antenna was horizontal and the AM transmitter tower was vertical, but the long wire antenna still picked it up quite well.