Abraham was the first Jew.  What was his original ethnicity?  He came from Ur.  Was he an Arab?  In any case, the genealogists figure that the mitochondrial DNA for
Abraham and his immediate descendants was J1c2b.  This would be from what we today call Israel or from the surrounding territories (I have not read a precise definition
of what "surrounding territories" means, but it would probably mean at least the lands through which the Euphrates river flows).  It would be a good study to find out whether
Abraham was of what we call Arab stock, or was he Turkic, or did he belong to yet another sub-group of the region, but I am told by a current genetics firm that the maternal
haplogroup J1c2b is from that whole area immediately around present day Israel, and including present day israel.

So by Paul the Apostle's day, many other strains came into the Jewish ethnicity.  Some were Egyptians, for instance, who joined the Jews and came back to Judea from
Egypt during the Exodus.  There were probably many other ethnicities in their genetic mix, acquired while in Egypt.  They were circumcised, if male, or were women who married
Jews and accepted the God of Abraham, Isaac, and Jacob.

Then after the Exodus, other nearby peoples became entwined in the Jewish DNA.   These would include converts such as Ruth and Rehab, but these women may have had
the ethnic DNA J1cwb too, coming from those who lived very close to Israel, indeed within walking distance.

Did the Jew intermarry during the Babylonian exile?  A good question, one I can't answer.  It would be interesting to have the opinion of a learned rabbi on this.  Certainly by
the time of the exile they may have been practicing exclusion of other tribes into their gene pool.  For long periods of their history, Jews did largely exclude other ethnicities
from marrying into their tribes, and this was reinforced during the period we know as the European ascendancy (middle ages, etc.)  During these long periods of exclusivity,
the ethnicity of the Jews became more established.  But still, to find a pure ethnic descent of any Jew is rare.   The haredi Jews (ultra Orthodox in Israel, and the same group
in, say, New York City under the late Rabbi Schneerson) claim that there are only three families who have maintained a direct physical descent from King David.  Needless to
say, these families are honored among the haredi Jews, and Rabbi Schneerson was a descendant of this.  

Paul tells us that a true Jew is of the "heart", not of the circumcision or birthright.  Anyone who accepted as Lord the God of Abraham, Isaac, and Jacob is a Jew, according to Paul,
but of course Paul would also include gentiles in this group, the uncircumcised who accepted Paul's God through accepting Yeshua as Messiah.  One could argue against this
being Paul's precise definition of who is a Jew, but to do this correctly one would have to look at each separate Jewish generation and see what their physical descent (or not)
from Abraham was.  Paul never intended us to do this impossible task of separation.