The main barrier was that, if you were a Latin Christian living in Europe, it would be a serious crime, legally speaking, and socially it would be just about one of the worst things you could do to yourself and your friends and family.

Apparently it did happen sometimes in the earlier medieval period. Once the Roman Empire was Christianized, Jews were clearly second-class citizens at best, and this was true of the Jewish communities in places that ended up outside the Empire, in the Germanic kingdoms in Italy, France, and Spain. The Empire legislated against it in the 5th and 6th centuries. Jews weren’t allowed to marry Christians; but if they did, and had children, the children were legally required to be raised as Christians. Jews couldn’t own Christian slaves; if an enslaved Christian was pressured to convert to Judaism, they would be legally free with no compensation to the owner. Jews couldn’t serve on municipal councils or as military commanders. In short they weren’t allowed to do anything that would place them in a position of authority over Christians.

By the time you’re asking about, many centuries later in 1250, Jewish communities still existed in the old cities of the empire, and in new cities - anywhere there was a large urban population, there was probably a Jewish community too. But they had sort of retreated into quasi-autonomous neighbourhoods. Jewish citizens weren’t allowed to do certain jobs, and they were subject to occasional persecutions, expulsions, and/or massacres. In France there had recently been attacks on the Jews in 1236, and all of the Jews were expelled from the Duchy of Brittany in 1240. That same year, the “Disputation of Paris” was held - essentially, the Latin church had discovered the Talmud, and learned that there are things in the Talmud that (from the Latin Christian point of view) could be considered heretical. The church challenged the Jews of Paris to a debate, which the Jews of course had no hope of winning, and in the end all copies of the Talmud in Paris were burned.

The church actually considered the Jews theologically necessary - they were evidence of the truth of the Old Testament, but also evidence that the New Testament was the fulfillment of the Old. Jews still had to exist because of Biblical prophecies that they would all convert at the end of the world (which was always about to happen…any day now!). Christians were now God’s chosen people, and obviously the Jews were miserable and had a low status in society because they rejected the truth of Christianity. The church was responsible for protecting the Jews, as if they were little children or orphans.

The only advantage that the Jews had, as far as Latin Christians were concerned, was that they appeared to be fairly rich - but of course that was only because there were so few avenues open to them. One of the things they were allowed to do, which Christians were generally not, was trade and lend money with interest. So some Jews could become relatively wealthy. But it was easy for Christian rulers to confiscate all that money and property and expel them from the country - as mentioned, this happened in Brittany in 1240, and it happened in England 1290, in France several times int he 13th and 14th centuries, and Spain in 1492. The First Crusade in 1096 started off attacking the Jewish communities along the Rhine, partly because they believed they could extort a lot of money from them to pay for the crusade.

So if you were an average Christian in Latin Europe in 1250, as far as you knew the Jews lived in insular communities. If you didn’t live in a city, you might never even meet any Jewish people at all. In your lifetime you had seen or heard about numerous attacks on Jewish communities, or you saw them expelled from nearby lands. You probably thought they were a bit strange and untrustworthy. And you may have been a bit paranoid that they were going to somehow trick people into converting...

Abandoning Christianity was known as apostasy. While the church in 1250 was probably much more concerned with Christians converting to Islam, apostasy to Judaism (also called “Judaizing”) was also a possibility. Someone who converted to Judaism would typically be a persona non grata in the Christian community. Their land could also be confiscated and they wouldn’t be allowed to inherit anything from their family. Children could also disown their apostate parents. Apostasy could even be punished by death.

The church built up a legal framework to punish people who apostatized from Christianity. The canon law of the church included decrees against conversion from as far back as the Council of Toledo in the 7th century. In 1267 Pope Clement IV issued “Turbato corde”, which declared that Christians who converted to Judaism should be treated as heretics - that is, they could be brought back into the church if they repented, but if they were “obstinate” and continued with their heresy, they could be executed. Secular law codes also contained similar instructions. Converts were to be executed in the Spanish “Siete Partidas” law code, which was also published in the 1260s.

The church was also afraid of the opposite situation, that Jews who converted to Christianity (or even Jews who had been baptized against their will) would secretly return to Judaism or continue practising Jewish customs (i.e. they would be crypto-Jews). That’s really a separate issue, but it was a major concern in 15th century Spain (and later).

Ok, so now that we know why it would be extremely unusual and dangerous…but what if you actually did want to convert? Maybe you questioned Christian doctrines, maybe you were attracted to the languages and literature of Judaism, or for whatever other reason, you felt that being Jewish was advantageous for you in some way. Or maybe you fell in love with a Jewish person and you wanted to convert to be with them.

Probably the most famous convert was Obadyah “the Norman” (or “the Proselyte”), who was an Italian nobleman or possibly a cleric, who converted to Judaism in 1102. Unfortunately we don’t really know why he converted. We don’t even know his original name (Obadyah was the Hebrew name he adopted), but it may have something to do with the First Crusade just a few years earlier. The Normans of Italy were heavily involved in the crusade, and as mentioned, the Jewish communities of Europe were often the first targets of the crusaders. Obadyah left a lot of writings, mostly about Jewish music, but he never explained his reasons for converting.

Closer to the time period you’re asking about, in 1222 a priest in Oxford fell in love with a Jewish woman. He gave up the church and his Christian faith, converted to Judaism, and even got circumcised in order to be with her. Unlike Obadyah, who lived the rest of his life in peace, this unnamed priest was caught and brought before a church council convened by the Archbishop of Canterbury, Stephen Langton. He was convicted of apostasy and executed.

We don’t really have any idea how many people converted to Judaism, but apparently the answer was “enough that the church was worried about it.” And unfortunately, we only really know about specific people who converted when they were caught and put on trial, or when they encountered the Roman or Spanish inquisitions, which were just beginning around your time period in the mid-13th century. So, it would be very unusual if you wanted to convert, but if you actually did, you'd be cut off from your family and Christian society for the rest of your life. And you'd have to make sure the church never came looking for you, because if they caught you, you could be killed.