Such a bridge would need to be around 35 kilometres long (assuming it spanned the Channel at its narrowest point). There are fourteen bridges in the world longer than 35km. Seven are in China, two in the USA and one in Thailand. Five of them are road bridges, one a combined road and rail bridge and the rest rail bridges. The longest is 164km in length – almost five times the length of a bridge across the Channel would need to be. So the technology to build ultra-long bridges certainly exists.
However, the majority of these bridges are low-level affairs, more like glorified viaducts and some involve considerable lengths of causeway rather than true bridges. Neither of these designs would be suitable for the Channel. Furthermore, most of the bridges span lakes or lake systems. Those that span open water are usually bridge-tunnel constructions where the road or railway is taken beneath the water for lengthy stretches to allow navigation.
Whilst it may be possible to design an acceptable structure its vulnerability would be considerable and I don’t believe this idea will ever gain much support.