How to access the default ASP.NET/IIS Error Descriptions programmatically?

As suggested in the comments, these strings are resources within the System.Web assembly, and there's no systematic means of retrieving just the ones related to HTTP error codes¹ (versus all of the other string resources within System.Web)

For example, this Console application, with a reference to System.Web:

using System;
using System.Resources;

namespace PlayAreaCSCon
{
    class Program
    {
        static void Main(string[] args)
        {
            var rm = new ResourceManager("System.Web",
                typeof(System.Web.HttpApplication).Assembly); //Arbitrary type
            Console.WriteLine(rm.GetString("NotFound_Http_404"));
            Console.ReadKey();
        }
    }
}

Generates this output²:

HTTP 404. The resource you are looking for (or one of its dependencies) could have been removed, had its name changed, or is temporarily unavailable. Please review the following URL and make sure that it is spelled correctly.

Note also that the fact that these strings are stored as resources is an implementation detail and not documented anywhere, so you'd have to accept that any code based on accessing these resources could be broken by any future updates to the .NET Framework (although it's unlikely to change in practice).

¹By which I mean I had to use a decompiler to find the magic string "NotFound_Http_404" used in this sample. There's no logical way to go from that to the needed string for any other status code.

²On a machine where no effort has been made to select anything other than the en-US locale. YMMV.