I would like to ask if it is possible to bezel set a flat sided
cabochon without using anything but the bezel itself.
It’s possible, of course, but might not be secure. The setting you
describe means the bezel wall is vertical, with nothing actually
coming over the stone in any way, to trap it into the setting, as is
usually the case. So to do this, your stone would have to be a tight
friction fit, and that friction is all that would be holding it in.
If those sides were, for example, slightly rough, rather than smooth
polished, it’s perhaps possible to get the metal tight enough to it
that this might be enough. But I would not recommend it. Normally,
when stones simply press fit into a setting, some sort of adhesive
would also be used. Epoxy, super glue, etc. Usually, when one needs
to set a stone like you have, with vertical rather than bevelled in
sides, one runs the bezel all the way up the height of the stone,
plus a little. Then that top edge can be burnished down, so it can
at least slightly grip the upper edge of the stone. That can indeed
be secure if done well, but you don’t have much room for error…
Easier would be to have any decent lapidary (if you’re not equipped
to do it yourself), recut the sides of your stone to a slight bevel
so you have a shape that a bezel can actually hold down into the
setting. Another way, if you prefer the look of a vertical bezel, is
to grind a groove around the side of the stone, or perhaps just a
couple short grooves, either way, within the area that the bezel will
hide. The bezel’s top edge is fitted to just the top of that groove,
and when burnished down/in, it can grab into that groove, while still
fully hiding it, so it appears to be simply a straight sided stone
and bezel, but the groove the metal is holding onto is not apparent.
This is somewhat a variation on the way “invisible setting” is done.