Just because they say something doesn't necessarily mean that they speak true. Even the remainder of the ayah indicates that what they said was nothing more than their own wishful thinking, rather than actual fact.
What really matters in this case is understanding who "they" is referencing — based on the facts in OP, it's unlikely to be the Jews in general — and why "they" would say something that is so apparently untrue.
Luckily for everyone here, I don't have to start working out any elaborate conspiracy theories involving little-known deviant sects of Judaism that just happened to be in existence during the time of the prophet (which is too bad, I had a couple of good ones). Pulling back a bit in the Qur'an to get some context makes it quite clear who is actually speaking here:
2:109 Many of the People of the Scripture wish they could turn you back to disbelief after you have believed, out of envy from themselves [even] after the truth has become clear to them. So pardon and overlook until Allah delivers His command. Indeed, Allah is over all things competent.
"They" isn't referring to all of the Jews, it's not even referring to most of the Jews. And it's definitely not referring to the actual teachings of the Jewish Scriptures.
"They," in this case, is referring to a specific subset of the People of the Scripture (which includes both the Jews and the Christians). Those who, rather than going their own way with their own religion, actively wish to turn the Muslims away from Islam.
They are the ones who say "None will enter Paradise except one who is a Jew or a Christian." They may actually even believe it, or they may be lying just to win their point, I couldn't say. Even nowadays, there are plenty of people who play fast and loose with the facts in their own Scriptures (Muslims included), people who are less concerned with relaying the true message of God than they are about proving others wrong.
But their actions, and their claims, hardly reflect Judaism (or Christianity) as a whole.