Ok, let’s just get one thing out of the way here (in case the title wasn’t clear enough)…
There is no such thing as a ‘Jewish race’.
The concept of race in an of itself is a social construct that was created by human beings (not God) as a means to label and stratify society.
What does exist are shared strings of DNA among people who set roots and developed in various parts of the globe. That shared DNA often yields distinct physical traits among those that posses this shared DNA.
However, trying to apply the scientific laws of DNA to define Jewishness is highly problematic. Let’s approach this from two different angles here:
Scientific

The DNA possessed by any human being on this planet will be 99.9% similar to that of any other human being. That is what makes us humans (as opposed to being dogs or cats). The differences in that remaining 1% are what makes us different between each other. But since DNA is highly complex, these small differences make a big impact. That is why DNA testing companies can market the ability to reveal the ethnicity of your ancestors. And they are not lying. But what is not made as obviously clear (unless you read the fine print) is that the margin of error your results can be up to 10%.
DNA is very effective on a micro-scale in regards to being able to show if you are related to others. On a macro scale, the confidence level decreases and the only way that your ethnicity (or race) can be determined is by comparing your results to that of a reference population. As more and more people file DNA test results, the reference sample will change. Also, there is no piece of DNA that makes you ‘Black’ or ‘Asian’ or ‘White’. Rather people who take DNA tests self-identify — and then their own definition of what their ethnicity is becomes the label of the reference sample.
See the problem with that?
Spritiual
Now let’s head over the the complete other side of the spectrum and look at classic Jewish philosophy on what a Jews actually is. The very first Jew, Abraham, was not born as a Jew. He came to recognize and acknowledge the Creator and whoever joined with him and his family in belief and practice became Jews as well. When Moses led the descendants of Abraham out of Egypt, then the official canon of Jewish law was received and what made you a Jew and how Jews were supposed to behave became codified. Again, you have people who were definitely born as non-Jews (i.e. Zipporah, Moses’s wife) in close proximity to those who were and all living as Jews together.

Couple that of the belief that there is such a thing as a Jewish soul. In some opinions, people who are born as non-Jews who convert to Judaism always had a Jewish soul. The conversion process was just a vehicle through which that soul could emerge. Once that person converts, than they are no less Jewish than any other Jew. However, nothing about their DNA changed.
What Exactly Prompted This Post?
The Jerusalem Post recently ran an article on Rabbi Angela Buchdahl — and the comment section just went wild with slandering. Her Jewishness was slandered and picked apart in really appalling ways and it was just very disheartening for me to read through.
To those who prescribe to the idea that yichus (Jewish lineage) magically imparts superiority; than Ms. Buchdahl can absolutely claim that she has ‘Jewish DNA’ (and you already know why I’m putting that in quotes) because she has Jewish ancestors. She does not have an unbroken Jewish maternal line. However I’m not sure that if you believe in Jewish racial purity, why that matters anyway. Jewish racial purity is not a religious/Torah concept. That is more espoused/supported by the secular Jewish crowd.
To those who say that Ms. Buchdahl’s conversion to Judaism was not kosher (hence she is not Jewish), I would question what understanding these individuals have of the halachaic requirements of a conversion. Even if they do, how do they have the intimate knowledge of what her conversion entailed?
So dare I introduce the lingering specter of good old fashioned racism? I’ve actually seen it among several of my Jewish friends who have married and had offspring with others of Asian descent. People just can’t get past the fact that someone doesn’t ‘look’ Jewish. Well guess what. Tons of people do not “look” like the stereotypical phenotype that presents itself in a particular racial or ethnic group. And then so what if you do? Or is it even possible? What does a Jewish nose look like? Jewish hair? Jewish eye color? Jewish skin color? Where along the spectrum does any of that lie? What if you have a unbroken maternal Jewish line; but the past 10 generations have intermarried so you have 2% Jewish DNA. What if you have 3 Jewish Grandparents, and one Gentile grandparent — who just happens to be your maternal grandmother….what then?
What frustrates me personally is not that Jewishness is not easily defined. That is not a new issue and it has no easy answer. What frustrates me are those who set up no-win scenarios. So you can be 75% Jewish according to your DNA test. Non-Jewish mother…..fine, not a Jew! But then the same people who made the previous rejection will also say you may have a Jewish mother….but you have 80% African DNA and you don’t look like any Jews I know, so no…..sorry, not Jewish!
It’s all a very fruitless and non-productive topic to discuss about someone. For me, if someone identifies as a Jew….if they have a community that they are accepted into. Then who am I to come in and make any declarations whatsoever?

Re: Rabbi Angela Buchdahl
(1) Under normative Torah Judaism, a woman can’t be a rabbi. No matter how lyrically she sings or strums the guitar.
A rabbi isn’t hired for his proficiency as a musician or a vocalist. He is hired to be a teacher and a Halachic decisor. I don’t know if Ms. Buchdahl has yet mastered Tractate Kesubos which we’re now studying in Daf Yomi. But I sorta doubt it.
(2) Kol Isha, a rabbinical enactment, prohibits Jewish men from hearing women sing. Period.
You might want to pick up a copy of “Guardian of Eden, In Search of the Jewish Woman,” by Rabbi Yisroel Miller.
In part two, “Letters to a Jewish Feminist,” Rabbi Miller argues forcefully why women shouldn’t get aliyot or participate more fully in synagogue services, or, by extension, even be ordained:
“It is quite understandable,” he writes, “that many Jews are upset with the Jewish woman’s limited role in the synagogue, because most Jews believe that ‘Jewish observance’ and ‘synagogue’ are virtually synonymous.
“To most contemporary Jews, ‘religion” means attending synagogue services and more religion means more services, as a participant or even only a mere spectator, three times a year or every day.
“But the idea of ongoing authentic religious experience outside the synagogue is foreign to them, except perhaps for a Passover seder or an occasional funeral.
“The synagogue as an institution is both worthy and sacred; but in the totality of Torah, it is but one small ingredient, a rabbinically ordained addition to the Biblical commandments.”
The upshot? There are many, many ways in which Jewish women can lead a Torah-true life. They don’t have to strum the guitar during services, or seek “semikhah ” from Rabbi Avi Weiss.
Thank you for adding your perspective/information. This post was focused on the concept of there being a Jewish race – not so much the halachaic permissibility of a woman being a rabbi. As a traditionalist I don’t dispute what you’ve presented. But I also respect the decision of the various Jewish movements to certify their spiritual leaders on whatever terms they find acceptable to them. Thankfully we have a choice to align ourselves with the community that supports our own beliefs. Thank you again for commenting!