This is actually not as silly a question as it might sound at first.
A “Roman” was anyone granted Roman citizenship—much as an “American” today is anyone granted American citizenship. Many foreigners (which, as far as native Romans were concerned, included all Italians not from Rome or its surrounding Latin region, as well as those from far off lands) were granted this cherished status, as a reward for service or for some other worthy merit.
Thus, even a Judean could become a Roman citizen. Ben Hur is a romanticised, fictitious character, but there were notable historical figures who were Judean and Roman:was one (technically), and so was Yosef Ben-Matityahu, a.k.a. .
To become a “Judean” (the word “Jew” really applies to people of Judaic faith after the destruction of Judea in 132 CE) if you were not born to a Judean mother was much more difficult—in some respects, well-nigh impossible. In much earlier times—say, the eleventh century BCE—one could become a Judean by simply declaring an oath of allegiance, as Ruth the Moabitess (great-grandmother of King David) did:
Where you go I will go, and where you stay I will stay. Your people will be my people and your God my God. Where you die I will die, and there I will be buried. May the Lord deal with me, be it ever so severely, if even death separates you and me.
But the start of the Common Era, which is the time of the Ben Hur story, it was much more complicated. Rabbi Hillel told one interested pagan who told him that would convert to Judaism if Hillel could teach him the whole of the Torah “while he stood on one foot.” Rabbi Hillel replied: “What is hateful to yourself, do not do to your fellow man. That is the whole Torah; the rest is just commentary. Go and study it.” (Talmud Shabbat 31a). Rabbi Akiva agreed, arguing that the essence of the Torah was “Love thy neighbour as thyself” (so now you know where Jesus got the idea). But other rabbis, such those of the stricter, Rabbi Shamai school of thought, demanded a more rigorous conversion process, to ensure that the convert was sincere. Even that, however, was not enough for some. Many Judeans refused to recognise King Herod himself as a true Judean, because he was of Edomite ancestry.