A plasma display has comparable picture to a CRT (very similar if not identical method of actually generating light, just different phosphor stimulation) and would be thin, but the smallest plasma you're likely to find these days is a 42".
CRT projection sets can be made reasonably thin, but there's no reason to do that. These days as you'd just use an LCD or DLP projector (I have no idea what happened to LCOS and SXRD).
I imagine it would be possible to build a CRT (physical tube) with a sort of "electron mirror" in it that would attempt to reflect the swept electron stream onto a CRT face much like is done with light in rear projection sets. That might be how that weird RCA CRT qrz linked works. I would guess it would suffer from very poor geometry. The reflecting surface also won't be anywhere near as efficient as a mirror is at reflecting light, so lots of excess heat would probably result. All in all, not practical, especially since I'm guessing you're not one of those rare breeds that enjoys making their own tubes.
The "flat screen" CRTs are just that: the front of the screen is flat. They didn't mention a thing about the depth of the whole set

Popular trade terms for such tubes are "PureFlat", "DynaFlat", "TrueFlat", etc. There also exist "Flat" tubes that aren't perfectly flat but are a lot closer than most. I have one such tube on my Kortek KT-2914F (not to be confused with the KT-2914DF which has a "DynaFlat" tube and was more popular). The "perfectly flat" tubes actually sometimes result in slightly deeper sets due to the odd angles and weird scanning waveforms required to get the geometry right.
If you've got room, you can use a mirror with a CRT below it as qrz suggested. Some older cabinets were made this way. It's still not quite like staring directly at a CRT face, and it requires some depth, just less than a normal mount. It also requires room in the bottom of the cabinet for the monitor.