Neither is timeless.
Emily was definitely old-fashioned when I was young (I had a doll in an Edwardian-type outfit and chose the name Emily deliberately to fit with that). Sarah was a pretty standard one-in-each-class kind of name. Much less popular now, though each of my sons (born 99 and 03) had a Sarah in his school friendship group.
It really doesn't matter. None of the kids at school will know that your child has the same name as thousands of 30-somethings.
I knew children called Leigh, Bethany, Abigail, Tricia, Annette, Roisin, Miriam, Anu, Kenneth, Roderick. (As well as all of us who were Julie, Susan, Andrew, Peter, Matthew, Donna, Rachel, Sharon, Tracy, Lorraine, Simon, Robert, Phil, Chris) I don't recall an unusual name ever being an issue.