"Messianic Judaism" is not a single denomination or sect; the term covers a broad range of beliefs and practices. Various groups exist; often the disagreements concern the extent toward which the Mosaic law must be followed today. I'm going to answer based on the views of two groups: the Messianic Jewish Alliance of America (MJAA), and the Union of Messianic Jewish Congregations (UMJC). 40-60% of Messianic Judaism's congregations are Gentiles, so this is an important question.
The Messianic Jewish Alliance of America
I'm going to sacrifice accuracy a bit to be concise: The MJAA tries to balance between Christianity and Judaism; if forced to distance themselves from one they tend to align themselves with evangelical Christianity.
From the MJAA Statement of Faith:
We recognize that Jewish people (physical descendants of Abraham through Isaac and Jacob, whether through the mother's or the father's blood-line) who place their faith in Israel's Messiah, Yeshua, continue to be Jewish according to the Scriptures (Romans 2:28-29). Gentiles who place their faith in Yeshua, are "grafted into" the Jewish olive tree of faith (Romans 11:17-25) becoming spiritual sons and daughters of Abraham (Galatians 3:28-29).
We acknowledge that the New Covenant body of believers is composed of both Jews and Gentiles who have received Yeshua the Messiah as the Promised Redeemer. The "middle wall of partition" has been broken down and now we worship the God of Israel together (I Corinthians 12:13; Ephesians 2:13-14).
Concerning the acceptance of non-Jews into their congregations ("Messianic Gentiles"), Kesher Journal reports (emphasis mine):
Congregations are faced with internal division from having a membership that includes Jews, who take part in the destiny and specific promises to the Jewish people, and Gentiles, who do not. The official stance is that Gentiles and Jews are spiritually equal but distinct, and that Jews should be proud of being Jews, and Gentiles proud of being Gentiles. Nevertheless, the Jewish identity is clearly valorized, causing many Gentiles to strive for greater Jewishness through Jewish observance and search for Jewish roots. Since conversion for Gentiles is deemed unbiblical within the MJAA, these are the main options for Gentiles seeking a more Jewish identity. What helps increase a sense of unity between Jews and Gentiles within these rituals is the stress on the importance of being a "spiritual" Jew more than a "physical" Jew...
Messianic Jews within the MJAA will accept a Gentile into their congregations, but not allow conversion to Judaism; you will be a "Messianic Gentile." As Ellen Kavanaugh puts it concerning Christians who consider converting to Messianic Judaism,
Conversion implies a change in belief -- from one belief system to a different belief system. And therein lies the problem: To convert to Messianic Judaism would be to 'convert' to a community in which one was already [a] member; to convert to traditional Judaism would be to deny one's faith in Yeshua (or at least, deny the sufficiency of Yeshua alone).
The Union of Messianic Jewish Congregations
The UMJC strives to be an authentic Jewish group, and be recognized as such by non-Messianic Judaism. They strive for "authentic Judaism."
They define Messianic Judaism as such:
Jewish life is life in a concrete, historical community. Thus, Messianic Jewish groups must be fully part of the Jewish people, sharing its history and its covenantal responsibility as a people chosen by God. At the same time, faith in Yeshua also has a crucial communal dimension. This faith unites the Messianic Jewish community and the Christian Church, which is the assembly of the faithful from the nations who are joined to Israel through the Messiah. Together the Messianic Jewish community and the Christian Church constitute the ekklesia, the one Body of Messiah, a community of Jews and Gentiles who in their ongoing distinction and mutual blessing anticipate the shalom of the world to come.
For a Messianic Jewish group 1) to fulfill the covenantal responsibility incumbent upon all Jews, 2) to bear witness to Yeshua within the people of Israel, and 3) to serve as an authentic and effective representative of the Jewish people within the body of Messiah, it must place a priority on integration with the wider Jewish world, while sustaining a vital corporate relationship with the Christian Church.
In the Messianic Jewish way of life, we seek to fulfill Israel's covenantal responsibility embodied in the Torah within a New Covenant context. Messianic Jewish halakhah is rooted in Scripture (Tanakh and the New Covenant writings), which is of unique sanctity and authority. It also draws upon Jewish tradition, especially those practices and concepts that have won near-universal acceptance by devout Jews through the centuries. Furthermore, as is common within Judaism, Messianic Judaism recognizes that halakhah is and must be dynamic, involving the application of the Torah to a wide variety of changing situations and circumstances.
Messianic Judaism embraces the fullness of New Covenant realities available through Yeshua, and seeks to express them in forms drawn from Jewish experience and accessible to Jewish people.
To again quote Kesher Journal:
In a recent discussion at the 2002 Lausanne Consultation on Jewish Evangelism (LCJE) Conference on Messianic Jewish Identity, a UMJC affiliated rabbi argued that Messianic Jews who do not keep kosher are in sin against God, denying their calling and identity... One of the implications of this stance is that observance of the traditions becomes a measure of spirituality and closeness with God, creating a spiritual hierarchy between those who are most observant relative to those who are less so. By keeping the law, Union congregations also hope to build bridges with observant parts of the Jewish community.
The UMJC, in contrast to the MJAA, requires "Messianic Gentiles" to observe the Jewish laws, and generally to convert fully -- becoming Messianic Jews.
From the Kesher Journal article once more:
In the past, Messianic congregations have generally defined themselves as a place where Jews and Gentiles worship together, witnessing to the unity of Jews and Gentiles. Many in the UMJC are seeing this as a defective definition. Congregations may have Gentiles, but they are not part of the definition of the congregation, which is to be a Jewish space. The congregation is not an adequate witness to the unity of Jews and Gentiles because the Gentiles there are called to live as Jews- the unity would only be demonstrated if Gentiles were to live as Gentiles.... The simple reality is that many Messianic Jewish leaders realize the kinds of congregations being built are unable to adequately express the Jewish life. They are not seen as authentically Jewish.
As with other issues, the UMJC's distinguishing mark is its concern for Jewish authenticity. One way to be more authentically Jewish is by adopting a conversion process, similar to other forms of Judaism.
In short, whether a Gentile can convert to Messianic Judaism depends in large part on the beliefs of the specific congregation to which he or she would like to join. In some congregations, you may be able to convert and become a Messianic Jew, while in others you will be accepted to the community, but as a "Messianic Gentile."