It entirely depends on the platform and representation.
For example, in .NET a string takes two bytes in memory per UTF-16 code point. However, surrogate pairs require two UTF-16 values for a full Unicode character in the range U+100000 to U+10FFFF. The in-memory form also has an overhead for the length of the string and possibly some padding, as well as the normal object overhead of a type pointer etc.
Now, when you write a string out to disk (or the network, etc) from .NET, you specify the encoding (with most classes defaulting to UTF-8). At that point, the size depends very much on the encoding. ASCII always takes a single byte per character, but is very limited (no accents etc); UTF-8 gives the full Unicode range with a variable encoding (all ASCII characters are represented in a single byte, but others take up more). UTF-32 always uses exactly 4 bytes for any Unicode character - the list goes on.
As you can see, it's not a simple topic. To work out how much space a string is going to take up you'll need to specify exactly what the situation is - whether it's an object in memory on some platform (and if so, which platform - potentially even down to the implementation and operating system settings), or whether it's a raw encoded form such as a text file, and if so using which encoding.