To truly answer the question you're asking, we'd have to have samples of ancient Israelite DNA. Unfortunately, we don't have much of that. (This is not as obvious as it sounds -- there are lots more samples of ancient European DNA, and even Neanderthal DNA.) In their absence, we need to do a bit of guesswork, and a bit of conventional history.

You may have heard of a group called the Samaritans. According to the group's own history, they represent a group of Jews who stayed in the Levant, while most other Jews were exiled in Babylon. They claim that Judaism changed in two major ways after the separation of these two communities:

1. The ancient Proto-Hebrew alphabet was replaced with a new one derived from square-form Aramaic.
2. The Oral Torah, also known as the Mishnah (half of the Talmud), was introduced, and became as important to the religion as the written Torah.

Due to the latter development, this new form of Judaism was known as "Rabbinic Judaism". This is the form practiced by virtually all Modern Jews. Samaritans, however, were left behind. To this day, they continue to use an alphabet based on Proto-Hebrew, and they reject the authority of the Oral Torah.

Samaritans are today a tiny, endogamous community. That, combined with their claims about their group origin, makes them an appealing target for genetic studies. And, in fact, studies suggest that Samaritans form a very tight genetic cluster, and are closely related to Kohanim, another group of Jews (the former Temple priests) who claim uninterrupted Levantine lineage. [1]

It's not quite what you asked, but I want to point out that modern-day Jews resemble the Samaritans a lot more closely than you'd think. Y-DNA analysis suggests that the male progenitors of most living Jews were from the same pool of Middle Eastern men.

Aside from sharing many Y-DNA lineages, autosomal admixture studies also suggest that most Jewish subgroups have anywhere from 40-70% Levantine/West Asian contribution to their DNA. Yemenite Jews may actually have more Levantine DNA than Samaritans. Druze, Maronites, and Palestinians are in the same ballpark as Samaritans. Other Jewish communities tend to have a bit more admixture, especially Ashkenazi Jews, but we're talking about a difference of maybe 15% -- one or two great-grandparents out of 8. Pretty much every Jewish person alive (barring converts) has the equivalent of one ancient Israelite grandparent out of 4. [2]

But again -- until we find more ancient Israelite DNA samples, it's hard to be absolutely certain about any of this. There have been major population migrations in the past. The only thing we can say for certain is that all of today's Jewish subgroups, including Samaritans, are closely related to each other, and to Palestinians and Druze and Maronites too.

[1] http://www.ncbi.nlm.nih.gov/pubmed/25079122

[2] http://bga101.blogspot.com/2013/10/eurogenes-k15-now-at-gedmatch.html

Question for the Historians: do Historians include genetics in their area of studies? Would Anthropologists be better equipped to answer?

The best resource I've found for this is Dionekes's blog where the author reports on genetic studies. For a more nuanced, complex answer to your question, you most likely want to search that blog for "Jew".

To sum things up, your question is very complicated. The first problem is genetic testing. Depending on what is being measured, you can have misleading results. A simple example would be a hypothetical individual whose mother's mother was Maori, and whose father's father was Japanese, and the other two grandparents were English. If we test his mtDNA (mitocondrial DNA - which comes from the mother), we'd reveal the person being Maori. If we test his YDNA (Y chromosome, which comes from his father), we'd find that he was Japanese. If we did a more wide ranging DNA test on other chromosomes, we may detect the mixture of Japanese, Maori, and English DNA. In history, it's not unthinkable to imagine a scenario where mostly women or mostly men would be the ones to pass DNA down to a descendant population.

This leads us into the second problem: It's possible that some Jewish groups went through a narrow founder effect, where the founder's DNA pool was different than the DNA pool at large. Imagine there are four variants of one gene in a population - spades, clubs, diamonds and hearts. The entire population has a 25% chance of having any of those genes (we'll say this is on the Y chromosome to keep it simple). But just like drawing four cards from a deck, it's possible to come up with just two spades, a heart and a club. Or any other possibility. If those individuals are the ones who survive and reproduce, carrying on those genes, while the population that remains gets wiped out, then trying to determine the founding population's gene pool is difficult. Even in a large population, the gene pool can drift over time, as various genes are less likely to be reproduced due to chance.

Most of what we can hypothesize about what the ancient Hebrew DNA pool looked like comes from testing modern Jews and trying to figure out what they share in common (hence problems with founder effects and genetic drift). There are some genetic testing of ancient remains, but as far as I know, they are of a few individuals, which may be misleading due to the nature of the sampling and the small number of individuals tested.

With all those caveats in mind when it comes to determining what the ancient Hebrew gene pool looked like, I'll state that the DNA studies we have now are mostly consistent with a large part of the Palestinian population being genetically similar to what the Hebrew DNA pool may look like (with varying degrees of admixture with other groups for Palestinians). In addition, the DNA studies are consistent with the ancient Hebrews being related to other Semitic people in the region, people whose descendants still live in the region.

So most likely, the answer is "yes". More specifically, modern Jews tend to show a shared genetics with Palestinians. But this can be a politically loaded question, especially when it comes from who are the descendants of the ancient Jewish population in what is now Israel/Palestine. Scientifically, it's also a question without a concrete answer - we can say the DNA evidence is consistent with a hypothesis, and that it's the most likely scenario and supported by historical evidence, but we can't say exactly what happened.