Specifying size of enum type in C

You can force it to be at least a certain size by defining an appropriate value. For example, if you want your enum to be stored as the same size as an int, even though all the values would fit in a char, you can do something like this:

typedef enum {
    firstValue = 1,
    secondValue = 2,

    Internal_ForceMyEnumIntSize = MAX_INT
} MyEnum;

Note, however, that the behavior can be dependent on the implementation.

As you note, passing such a value to a function will cause it to be expanded to an int anyway, but if you are using your type in an array or a struct, then the size will matter. If you really care about element sizes, you should really use types like int8_t, int32_t, etc.

I believe there is a flag if you are using GCC.

-fshort-enums

Is there a way to tell your compiler specifically how wide you want your enum to be?

In general case no. Not in standard C.

Would it even be worth doing?

It depends on the context. If you are talking about passing parameters to functions, then no, it is not worth doing (see below). If it is about saving memory when building aggregates from enum types, then it might be worth doing. However, in C you can simply use a suitably-sized integer type instead of enum type in aggregates. In C (as opposed to C++) enum types and integer types are almost always interchangeable.

When the enum value is passed to a function, will it be passed as an int-sized value regardless?

Many (most) compilers these days pass all parameters as values of natural word size for the given hardware platform. For example, on a 64-bit platform many compilers will pass all parameters as 64-bit values, regardless of their actual size, even if type int has 32 bits in it on that platform (so, it is not generally passed as "int-sized" value on such a platform). For this reason, it makes no sense to try to optimize enum sizes for parameter passing purposes.

There is also another way if the enum is part of a structure:

enum whatever { a,b,c,d };

struct something {
   char  :0;
   enum whatever field:CHAR_BIT;
   char  :0;
};

The :0; can be omitted if the enum field is surrounded by normal fields. If there's another bitfield before, the :0 will force byte alignement to the next byte for the field following it.