They look like very nice pearls, but they don't look like Mikimoto to me because the clasp isn't a Miki clasp. The box is old and may have been reused.
The nice handkerchief in the bottom is exactly the sort of thing my mom and grandmother did to worn out boxes, when they wanted to keep something nice in there.
I wonder if the pearls were bought in Japan, because they are strung without knots, which is common there.