Not sure how Transport Canada views this but south of the border you can rebuild/restore an aircraft without logs or even an original data plate in some cases. The main issue is what type of airworthiness certificate you want. Unless you get a letter from the current Type Certificate holder of the model you rebuild the best you could hope for would be a restricted or experimental cert instead of a standard cert.
As for cost, I'd say if this were a Cessna 100 series or a Super Cub it would be doable, but a Beaver would be prohibitive, in my opinion. However, if you start with a decent core, like a weather damaged aircraft, and have in-house talent, then it may work. But its a big risk still. If your plan is to drop off logs and plate at the local airport shop and turnkey entire build then I would definitely say no way Jose. W1