[guest39538]

A man looks at an object in the distance using his eyes to judge the distance

The man then looked through a pair of binoculars to see the object appeared much closer

The man then by accident used the binoculars the wrong way around to see the object appeared much further way.

The man sat and thought

Which distance observed was telling the truth?

[alancalverd]

We don't observe distance. We infer it from our binocular angle. A one-eyed man just sees the object subtending a larger or smaller angle in his field of vision, but has no clue as to distance, which is why we go to the expense and complication of making binoculars, which are much more complicated than just "two telescopes".

[evan_au]

the object appeared much closer.. the object appeared much further way

For well-known objects of fixed size, the subtended angle gives a good indication of distance.

It is much harder to sense the distance of unknown objects.

Often people sense the distance between objects by noticing that the different objects require a different focus. For the naked eye, this only works for very small distances (<0.5m), but the effect is much more pronounced for large binoculars or telescopes.

Which distance observed was telling the truth?

The correct distance is determined by parallax - what different angle does the object present against the background.
Your eyes are pretty useless at determining the parallax beyond about 5m distance.

Small opera glasses have the same separation as your eyes, but also have some magnification, so you may be able to see a small shift against the background as you alternately open and close your eyes.

Bigger binoculars with bigger lenses have a "dogleg" in the optical path, which moves the center of the lenses farther apart than human eyes - this accentuates the parallax baseline, and the greater magnification lets you see smaller movements against distant objects.

A one-eyed man .. has no clue as to distance

Parallax does not work from a single eye. A one-eyed man can get a sense of depth by moving his head.

2D TV or cinema can give an impression of depth by moving the camera across the line of sight to the object, forming an artificial parallax.   

"One-eyed" astronomical telescopes (eg the Gaia satellite) use the motion of the Earth around the Sun to get a very large parallax baseline, allowing them to measure distances in our galactic neighborhood very accurately.

[alancalverd]

There have been a few one-eyed professional batsmen and goalkeepers, but as the greatest players say, at the top level it's as much a matter of learned reflex and anticipation as conscious image processing, and all those I can recall, lost an eye after several years of international sport.