Mishnah 2. The Mishnah establishes that a Jew may turn to a non-Jew for the healing of his property - that is, the healing of his animal, like a veterinarian - but not for the healing of his own body. A Jew may therefore go to a non-Jewish veterinarian, but not to a non-Jewish doctor. The concern is that the healer will act against the Jew's interests, make him ill, and even kill him.
The leniency in our times:
Today there are many grounds for leniency here, since there are consequences to a doctor's actions: someone who functions as a doctor and kills his patient - this is bad for him and for his livelihood, and it may even land him in police custody. Because these protections exist, along with natural economic deterrents, today one may turn to anyone who is called a doctor. The Mishnah, by contrast, assumes a wild, unsupervised, lawless society, in which the doctor could not be trusted to hand over a medicine.
It is worth noting that even in those days the healer could be relied upon for advice alone, since presumably the person asking would also seek a second opinion, and if the advice turned out to be bad it would damage the healer's livelihood. The Mishnah's sense, then, is that the danger lies in the healer being the actual killer.
A haircut from a non-Jewish barber:
Rabbi Meir: When the barber uses a blade, it is forbidden in every case - even in public, in a public barbershop where other customers are present. Once the non-Jew is holding the Jew as he pleases, with his neck exposed and the blade in hand, he will always find a moment when nobody is looking, and he will stage an "accident" - and slit his throat.
The Sages: They are somewhat more lenient - in a public setting such as a barbershop it is permitted, since people are around and a barber does not kill in front of people; but in a private setting it is forbidden.
And if you ask: we have already learned that one may not be alone with a non-Jew, and all the more so that one should not hand over his neck to a non-Jewish, idol-worshipping barber holding a blade - what does this ruling come to teach us?
The answer is that we are dealing with a private place - for example, a barbershop located in the barber's home, where people come and go, but where there may be moments when the barber is alone with the Jew. Under ordinary circumstances such a scenario would be permitted, because it does not qualify as being secluded together: being together for just a moment, like riding in an elevator, is not seclusion. Here, however, since one second and a slight flick of the wrist is all it takes, it is forbidden even in this scenario.
The practical halachah:
Here too there is ample room to be lenient, because of the consequences barbers face. Moreover, there is a general point we have already noted: if the barber uses scissors rather than a razor, it is permitted, since scissors are not as dangerous as a razor in terms of the speed with which the barber could kill the Jew, God forbid.