From a bridge engineering point of view (an area I’ve worked in for 25 years) it is technically possible but would be prohibitively expensive. I’m not sure of the depth of the Channel but you would need piers going to the sea bed which would be expensive and time consuming to construct. It would be a multi-span bridge so you would need a lot of piers which would be very expensive. You would probably have to do something like they did on the Øresund crossing and build an artificial island or maybe 2 due to the distance to be spanned. You would end up with a structure similar to this one in China.
http://www.bbc.co.uk/news/world-asia-pacific-13976281
Then comes the issue of shipping. A bridge over the channel would funnel down shipping to narrower areas which I would imagine would be a problem.