I would say your clutch is not releasing properly.  Could be a busted
section of the clutch disk, or the clutch cylinders are not doing their
thing.

> With the engine
> OFF, I was able to put it into 1st, start the engine, and hobble home in
> 1st.  If I take it out of gear now with the engine on, it's almost
> impossible to get it back into any gear.
>
> AND (I think this is important), depressing the clutch pedal doesn't seem
> to take the transmission out of gear (this happens in 1st and reverse).