Can a Jewish person believe in Jesus and still be a Jew? Of course! That is like saying a person who is Arab or Chinese is now a non-Arab or non-Chinese because they believe in Jesus.
Maybe the question ought to be restated by defining what it means to be a Jew. Actually the word is derived from a biblical reference for those who were the natural descendants of Judah yet today many have classified a Jew based on human opinion.
One opinion states that you are Jewish only if your mother is a Jew. Some claim that it is merely a matter of religion. Then how can the atheist, humanist, or reformed Jew whom the Orthodox don’t even recognize and who do not acknowledge the Torah as being the word of G-d be Jewish? Maybe its a matter of ethics? Then how about a corrupt Orthodox rabbi? Is he still considered a Jew? Is being a Jew a matter of showing solidarity with the people of Israel? Then what about the anti-nationalist Israelis? Perhaps its only ones ethnicity in which none of these other points are no longer important. Whatever the viewpoint is when denying someone an identity based on ones own criteria as listed above than this can be bigoted and who knows maybe by someone else’s definition they might even consider you to be outside of whatever group you identify with.
Anyway in getting back to the original question on whether you can be a Jew and believe in Jesus even the Orthodox recognize Jewish believers in Jesus as being “apostate Jews” and target them for outreach purposes.
The original context of what it means to be a Jew by biblical definition is one who is a descendant of Judah. If you look at the two greatest men that the Jewish people recognize and identify with as being their own then perhaps you might be interested in knowing that Abraham was neither a Jew nor an Israelite. Moses was an Israelite but not a Jew. Even among those who could make claims to being Jewish by biblical definition as descending from Judah can not prove the purity of their ethnic line because of the Diaspora, mixing of tribes, and converts to Judaism. So as you can see being a Jew isn’t always so definable.
Today there are many contemporary opinions and definitions about what it means to be a Jew. The opinions may vary and may fluctuate yet lets not try to limit a definition by excluding these Jewish believers in Jesus. Many of them still identify with the Jewish community yet their greatest identification is in their Jewish Messiah Jesus(Yeshua). These Jewish believers relate to the G-d of the Jews/Israel based on being a new covenant(testament) people as defined by the prophet Jeremiah as found in the Tanakh, Jeremiah 31:31-34.
31 The time is coming,” declares the Lord,
“when I will make a new covenant
with the house of Israel
and with the house of Judah.
32 It will not be like the covenant
I made with their forefathers
when I took them by the hand
to lead them out of Egypt,
because they broke my covenant,
though I was a husband to them,”
declares the Lord.
33 “This is the covenant I will make with the house of Israel
after that time,” declares the Lord.
“I will put my law in their minds
and write it on their hearts.
I will be their God,
and they will be my people.
34 No longer will a man teach his neighbor,
or a man his brother, saying, ‘Know the Lord,’
because they will all know me,
from the least of them to the greatest,”
declares the Lord.
“For I will forgive their wickedness
and will remember their sins no more.”
Lastly if we allow for every one else within the Jewish community to define what it means to be a Jew then those who are Jews who believe in Jesus should likewise have those same rights to identify themselves as being Jewish whether we accept it or not.
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