Why can't I have "public static const string S = "stuff"; in my Class?

A const object is always static.

From the C# language specification (PDF page 287 - or 300th page of the PDF):

Even though constants are considered static members, a constant declaration neither requires nor allows a static modifier.

A const member is considered static by the compiler, as well as implying constant value semantics, which means references to the constant might be compiled into the using code as the value of the constant member, instead of a reference to the member.

In other words, a const member containing the value 10, might get compiled into code that uses it as the number 10, instead of a reference to the const member.

This is different from a static readonly field, which will always be compiled as a reference to the field.

Note, this is pre-JIT. When the JIT'ter comes into play, it might compile both these into the target code as values.