Is period period period grammatically correct?

Answer 1

Yes, the three periods are called an ellipsis (...) which is a punctuation used to indicate that a part of a quote is missing. The ellipsis marks the place where the text is absent. For example:

Excerpt from MKL "I Have a Dream..." speech:

"It is obvious today that America has defaulted on this promissory note ..., America has given the Negro people ..., a check which has come back marked 'insufficient funds'."

Ellipsis are also used to indicate an unfinished thought or sentence:

"John said..." "What did John say?"

"John! Do not put lighter fluid on the hot..." "John, John are you OK?"

Answer 2

No, using "period period period" is not grammatically correct. "Period" is typically used to end a sentence or indicate a pause, so repeating it like that does not follow standard grammar rules.