Since Microsoft Web API isn't MVC, you cannot do something like this:
var a = Request.MapPath("~");
nor this
var b = Server.MapPath("~");
because these are under the System.Web namespace, not the System.Web.Http namespace. 
So how do you figure out the relative server path in Web API ?
  I used to do something like this in MVC:
var myFile = Request.MapPath("~/Content/pics/" + filename);
Which would give me the absolute path on disk:
"C:\inetpub\wwwroot\myWebFolder\Content\pics\mypic.jpg"