You certainly won't wind up with a white elephant but it might be a costly elephant. Buying all of the components individually and piecemeal can get expensive fast. Shop carefully and haunt the bike shops for NOS parts and bargains. E-bay can produce some great prices and some enormous headaches so use it with caution.

First, I'd read up on what's out there so you know what you want and how "modern" you want to go. Sheldon Brown's web site has a lot of good articles and maybe your local bike club has some experienced members that can give you help and advice.