Well, first of all from your map it appears that there are several rivers leading into cardigan bay. Now, unless you're including massive redirection of the rivers in this project, I would expect them to just fill the bay back up.
Also, the wall would have to be very thick. I think the great barrier dyke in the netherlands is something like 30 km long, and 90 m wide? While I do think that says it's possible, it would be difficult.
Economically I'm not sure what gould it would be. Farming would be difficult at first with the extremely high salt content in the soil. I'm not sure what the bedrock would be like, since I'm not an marine scientist. But if the bedrock IS there, I suppose you could build housing.