What you are looking to do is create a full email server. Here is a Code Project article that will show you how it can be done:
SMTP and POP3 Mail Server
Here is another site that might be useful:
In the end, however, you will need to create the website that displays the emails and allows the users to send email. Sending email via C# is not difficult. Receiving and displaying email will be more difficult.
As far as my recommendation, I would recommend against doing this. Trying to recreate an email server is not a good idea. There is a lot more that goes into them besides just sending and receiving email. I would recommend instead that you use an existing email server and just put your own front-end on it (if you really need to). Most email servers come with a web front-end, though, so it really isn't necessary.