Steve Willner <swillner@cfa.harvard.edu>

Very unlikely.  In the 1980's there was proposed a small companion, nicknamed
Nemesis, in a 26-million-year highly eccentric orbit, to explain apparent
periodicities in the fossil extinction record.  However, these periodicities
have turned out to be more imagined than real, so the driver for the existence
of Nemesis is gone.

Furthermore, such an object would be relatively close by, bright enough in the
infrared to have been detected easily by IRAS, and its high proper motion
should have been detected by astrometrists long ago.

One very slim possibility is that a very faint companion now located
near the aphelion of an eccentric orbit is not ruled out.  Such an
object would be hard to detect because its proper motion would be
small.  It's not clear, however, that an orbit consistent with the
lack of detection would be stable for the Sun's lifetime.

So the chances are that there exist no stellar companions to our Sun.