There are certain genetic markers that were found to be in common with most Jewish people (and some Samaritans) that are not found in other Gentile peoples. But the problem is that many of the Eastern Jews have a lot of Khazar blood, and don't look like the Western European Jews.
A VERY political subject, and one that almost most no one likes to look into. The Khazar were a Turkish people between the Eastern Roman Empire and the Caliph that decided to split the difference and become Jewish. There were already a large number of Jews living in the land, so this wasn't as surprising as it could have been. The claim that "All Jews in Israel are Khazars" is made by many islamics to say that Israel doesn't have a right to exist.
Why don’t they just start measuring skulls like the Nazis used to???
Correct.
I believe there was a thread on FR some time ago. The Jews claiming to be descendants of Aaron were sampled and compared to everyone else. They determined through some process that there was, indeed, a very high probability that the individual claiming such lineage was a valid priestly candidate.