Do women run out of eggs?

Yes. Yes, women do run out of eggs. As women approach menopause the ovarian reserve or supply declines. Certain medical conditions, medications, and treatments can decrease a woman's supply of eggs.
Yes. Each woman is normally born with a limited number of eggs, (typically about 100, 000). If everything is running smoothly, the eggs are depleted by the average age of 50. Without ovulation, or expulsion of eggs, the feedback mechanism to the brain to produce more estrogen is gone and estrogen stops being produced, causing the symptoms of menopause.