Best practice for constant string for implementations to use

I don’t think there is a best practice. It is just a matter of preference.

I store constants inside static classes.

public static class Constants
{
   public static class Messages
   {
      public const string Error = "Error accessing api...";
      public const string Hello = "Hello ...";
   }
}

Usage

var result = new TextResult(Constants.Messages.Error);

FYI: Some developers prefer Enum.