I'm sure this has come up before but I forgot the Terutz. The Gemara (daf 68) states that Rav and Shmuel both say our Mishne is relating to a ger who was converted amongst the nochrim. Such a ger will be chayav one chatas for all the Shabbatot and acts of chillul Shabbos that he incurred. Tosfos points out that there was an actual beis din that was megayair him or else he wouldn't be considered a ger.
This inyan seems to be a pele aztum to me. How can a person who didn't learn one of the very basics of Yiddishkeit, Shabbos, possibly be considered a ger? What kind of kosher beis din could have converted him? Didn't we learn that we must teach a ger miksat mitzvot kalot and miktzat mitzvot chamurot and he has to accept everything before he is eligible for gerut? I'm sure that no kosher bet din today would accept such a ger. Can the standards have been so lax in those days or is the whole inyan simply a construct? Might the Gemara be saying that such a ger is considered Jewish b'dieved if he accepted the mitzvot but didn't learn the basics?
A Kosher Gerus requires a Beis Din, but while it certainly should be made up of competent Rabbis who know what they're doing, it doesn't have to be. The minimum requirements of the Beis Din are that it be made up of three Torah-observant adult Jewish males who aren't Gerim themselves and who haven't done anything to invalidate themselves from being Dayanim. The Rambam (Hilchos Isurei Bi'ah 13:15) says that in the times of David ha'Melech and Shlomo ha'Melech, the official Beis Din stopped accepting Gerim because it was assumed that anyone interested in converting was just doing so to share in the power and prestige of the Jews of the time. However, groups of Hedyotos (laymen) formed Batei Dinim of their own and accepted these non-Jews as Gerim. The Rambam concludes that these conversions were post facto Kosher. The Rambam also says (ibid. 13:17) that even if the Beis Din didn't inform the prospective Ger of any of the Mitzvos, the conversion is also post facto Kosher. Hence, a Ger she'Nisgayer l'Bein ha'Nochrim is certainly a possibility.
Let me add that I don't think we have to infer anything about the standards of conversion in the times of Chazal based on the case of Ger she'Nisgayer l'Bein ha'Nochrim. The Gemara is simply discussing theoretically what would happen in such a case.
This is not a Psak Halachah