A quick search shows that Callisto has an apparent magntidue of 5.65, which would make it easily visible under relatively dark skies. Being the farthest Galilean moon, does it ever get far away from Jupiter for it to be discernable to the naked eye?
Is it possible to see Callisto with the naked eye when it's at its greatest elongation from Jupiter?
This has been an area of controversy for quite some time now, with no clear resolution (if you'll pardon the pun). There's been interest in supporting or refuting claims that observers were able to see the moons with the naked eye before Galileo. There are, of course, two major impediments (which you might well know about): brightness and angular separation from Jupiter.
The human eye has a diffraction-limited resolution of (roughly) one arcminute, whereas Callisto can reach separations from Jupiter of about 10 arcminutes - so, arguably, resolving it is not an issue, as it is with some other moons. However, we do have to deal with the problem that light from Jupiter could dominate any light we see from the moons (even if Ganymede and Callisto are too close to be separated and appear as one source). It's the same issue we get when observing faint sources near bright sources using a telescope, such as direct imaging of exoplanets.
If you do want to give this a shot, you might want to block out Jupiter's light with some foreground object, while making sure not to block out the moons. Choose a dark night when Callisto is at its greatest separation, and you might have a shot.
But not great odds, of course. I'd be surprised if you succeed. On the other hand, get yourself a good pair of binoculars and you won't have a problem.
1$\begingroup$ Might you see Callisto briefly if it was following Jupiter into occultation by the dark limb of the Moon? $\endgroup$ May 8, 2020 at 10:12
$\begingroup$ This shouldn't be all that hard to test. Look at Jupiter and Callisto through a telescope, note Callisto's position as seen thereby (in terms like "at two-o'clock"), and then ask a naked-eye observer who doesn't know the result to make the same observation. If they can reliably give the same answer, QED. $\endgroup$ May 8, 2020 at 14:45
3$\begingroup$ I expect that if the sky were clear enough to see Callisto, there would also be numerous low-magnitude stars visible. Good luck distinguishing the satellite from a background star. $\endgroup$– BarmarMay 8, 2020 at 18:15
$\begingroup$ @EvilSnack Well, there are plenty of problems - one of which is that the observer may not be reliable. It's also quite easy to confuse a moon with a background star or other object, as Barmar mentioned. Plus, there's confirmation bias to deal with. $\endgroup$– HDE 226868 ♦May 8, 2020 at 20:08
2$\begingroup$ @SteveLinton You don't need to wait for an occultation; Jupiter isn't bright enough to produce a lot of glare from atmospheric diffusion. Just block its light with a power line perpendicular to the plane of the moons. That's how I saw them as a teenager (probably after reading the cited Sky and Telescope article!) $\endgroup$– jeffBMay 8, 2020 at 23:32