It's possible, but that's not the usual construction. This is all about the grammar of "exclamatives"; here's how they work:
If you want to exclaim that the rain is very beautiful, then you should not invert the subject and the auxiliary, so you would say "How beautiful the rain is! Normally, inverting the subject and verb makes a question. Compare:
(1) How old he is! (exclamative)
(2) How old is he? (question)
Ex (1) is an exclamative; there is no subject-verb inversion. Exclamative "how" indicates a high degree - it comments on him being amazingly old. Ex (2) by contrast is a question; it has subject-verb inversion - it merely asks how old he is.
Exclamatives also occur with "what". Compare:
(3) What a car that was! (exclamative)
(4) What car was that? (question)
Ex (3) is an exclamative; there is no inversion. Exclamative "what" is concerned with quality or degree - it was a remarkable car. Ex (4) on the other hand is a question; it has subject-verb inversion; here "what" questions identity - the answer to the question will identify the relevant car.