All the members of Orion are within the Milky way, and some of them look pretty close to a neighbor. And could the bow* be an effect similar to the Radcliffe Wave?
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The three stars in Orion's belt, along with Sigma Orionis and the Orion Nebula cluster, along with several other young clusters and star forming regions in the central Orion region are all part of the Orion OB1 association at a distance of 350-420 pc.
A review of Orion OB1 can be found in Bally (2008). The three belt stars are probably a similar age and distance and form the brightest stars of the OB1b sub-association. Measured distances to these stars are individually rather uncertain, but better estimates come from the low-mass, co-moving stars around them. It is therefore true to say that the individual distances to each star are not known well enough to be sure that they are close together in distance.
The other bright stars in Orion - Betelgeuse, Rigel, Bellatrix, Saiph are significantly closer than this by 150-300 pc and probably not at a common distance. It has been speculated in the past though that these stars and the Orion OB1 association are part of one, large star forming complex, extending 200-300 pc along the line of sight (Bouy & Alves 2015).
The recent paper, claiming the discovery of a "Radcliffe wave" gas structure in the solar vicinity, does include the young stars in Orion as part of this structure.
So to spell it out: The belt stars are within 70pc of each other in terms of distance. If they are at a similar distance of 400pc, their angular separation on the sky of around 3 degrees, translates to a tangential distance separation of about 20 pc. Other members are more separated than this.