what is the size of an enum type data in C++?

This is a C++ interview test question not homework.

Then your interviewer needs to refresh his recollection with how the C++ standard works. And I quote:

For an enumeration whose underlying type is not fixed, the underlying type is an integral type that can represent all the enumerator values defined in the enumeration.

The whole "whose underlying type is not fixed" part is from C++11, but the rest is all standard C++98/03. In short, the sizeof(months_t) is not 4. It is not 2 either. It could be any of those. The standard does not say what size it should be; only that it should be big enough to fit any enumerator.

why the all size is 4 bytes ? not 12 x 4 = 48 bytes ?

Because enums are not variables. The members of an enum are not actual variables; they're just a semi-type-safe form of #define. They're a way of storing a number in a reader-friendly format. The compiler will transform all uses of an enumerator into the actual numerical value.

Enumerators are just another way of talking about a number. january is just shorthand for 0. And how much space does 0 take up? It depends on what you store it in.

The size is four bytes because the enum is stored as an int. With only 12 values, you really only need 4 bits, but 32 bit machines process 32 bit quantities more efficiently than smaller quantities.

0 0 0 0  January
0 0 0 1  February
0 0 1 0  March
0 0 1 1  April
0 1 0 0  May
0 1 0 1  June
0 1 1 0  July
0 1 1 1  August
1 0 0 0  September
1 0 0 1  October
1 0 1 0  November
1 0 1 1  December
1 1 0 0  ** unused **
1 1 0 1  ** unused **
1 1 1 0  ** unused **
1 1 1 1  ** unused **

Without enums, you might be tempted to use raw integers to represent the months. That would work and be efficient, but it would make your code hard to read. With enums, you get efficient storage and readability.

An enum is kind of like a typedef for the int type (kind of).

So the type you've defined there has 12 possible values, however a single variable only ever has one of those values.

Think of it this way, when you define an enum you're basically defining another way to assign an int value.

In the example you've provided, january is another way of saying 0, feb is another way of saying 1, etc until december is another way of saying 11.

It depends. The standard only demands that it is large enough to hold all values, so formally an enum like enum foo { zero, one, two }; needs to only be one byte large. However most implementations make those enums as large as ints (that's faster on modern hardware; moreover it's needed for compatibility with C where enums are basically glorified ints). Note however that C++ allows enums with initializers outside of the int range, and for those enums the size will of course also be larger. For example, if you have enum bar { a, b = 1LL << 35 }; then your enum will be larger than 32 bits (most likely 64 bits) even on a system with 32 bit ints (note that in C that enum would not be allowed).