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In this article it says:

The electrons can only orbit stably, without radiating, in certain orbits (called by Bohr the "stationary orbits") at a certain discrete set of distances from the nucleus. These orbits are associated with definite energies and are also called energy shells or energy levels. In these orbits, the electron's acceleration does not result in radiation and energy loss as required by classical electromagnetics. The Bohr model of an atom was based upon Planck's quantum theory of radiation.

My question is: how can this be if it is actually impossible? Any accelerating point charge should emit radiation, as explained here, for example. It is not like I can posit the existence of an orbit where the electron is stable, just out of mid air. What am I missing here?

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You are missing nothing. The Bohr model of the atom is false, and nowadays we replace the idea of the semi-classical "orbit" of Bohr with the fully quantum mechanical notion of orbitals or electron clouds, which give a probability distribution for the position of the electron around the nucleus, but do emphatically not imply that the electron is moving in any classical sense.

The reason the Bohr model was conceived of is that it can explain spectral emission/absorption lines like the Rydberg series by the energy difference between the orbits without being a full quantum theory, and it hails from a time when we did not yet understand quantum mechanics. Though its classical picture of orbits is completely false, its predictions for their energy levels are (modulo more subtle quantum effects) correct. Since, at the time, it was the first model to somewhat successfully explain the spectral lines, it has stayed around.

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    $\begingroup$ So, what was the point of even conceiving of such a model if it implies contradictions? $\endgroup$
    – user132181
    Commented Sep 25, 2014 at 13:31
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    $\begingroup$ @user132181 Because even with such contradiction was a useful model. It paved the way for later developments without such contradictions, also known as quantum mechanics. $\endgroup$
    – Ignacio Vergara Kausel
    Commented Sep 25, 2014 at 13:38
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    $\begingroup$ @user132181: I updated the answer. Basically, it was the best model we had back then, and for some reason it has lingered around. $\endgroup$
    – ACuriousMind ♦
    Commented Sep 25, 2014 at 13:38
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    $\begingroup$ My guess would be that, upon conceiving the model, Bohr simply rejected the idea from classical electrodynamics that accelerating particles radiate. So there was no contradiction; the old theory of electromagnetism was just considered incomplete, and indeed it is. $\endgroup$
    – gj255
    Commented Sep 25, 2014 at 13:45
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    $\begingroup$ All models are wrong in some aspect, but some of them are useful. $\endgroup$
    – Peteris
    Commented Sep 25, 2014 at 18:28
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The following passage has been extracted from Bohr's Nobel lecture:

While in contradiction to the classical electromagnetic theory no radiation takes place from the atom in the stationary states themselves, a process of transition between stationary states can be accompanied by the emission of electromagnetic radiation, which will have the same properties as that which would be sent out according to the classical theory from an electrified particle executing an harmonic vibration with constant frequency.

Bohr proposed this as an postulate (premise or starting point of reasoning). So, in a sense, he said this as a definition.

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