Andrew,
Just wanted to provide clarification on what "equal temperament" actually means for the piano. You are right that it's awful for strings.
Equal temperament does not involve shortening the octaves of the piano. You might be thinking of stretch tuning, which is the process of tuning higher strings sharper so that they sound more in tune with the harmonics of the lower strings. This is more of a problem with smaller pianos, and less of a problem with big concert grands.
In true equal temperament, if you were to play the C notes in every octave on a piano at once, they should sound perfectly in tune. Same for all the G notes, A notes, etc. What equal temperament does is divide the octave into 12 equal half steps. So this means that, in the key of C, there's an equal distance between the 3rd and 4th (E and F) as there is between the 7th and octave (B and C). The same is true for the distance between F# and G, and so on.
The problem for us fretless string players is that, in order to sound in tune, we have to shorten or lengthen certain half steps. For example, the 7th note of a major scale needs to be much sharper because our ears want it to resolve up to the tonic. Likewise, the 3rd will sound in tune when it's flatter, and the 5th when it's slightly sharper. However, when you're playing with a piano or guitar, it's not always possible to do this and still be in tune with the other instrument. Also, since most tuners are set to tune for equal temperament (100 cents per half step), you can be playing a 3rd that sounds perfectly in tune, but the tuner will tell you you're flat.
You can do a few things to work on your intonation. I think the best is to practice with a drone. If you have a tuner or a metronome that can sound a note, set it to the tonic of your piece and play against it. You'll be able to hear all the places where you are going out of tune, and you can practice correcting. Alternatively, if you have a tuner that can tune to the just intonation system, it can help to play against that.
Above all, intonation needs to be flexible! It depends on the context of the music, but also on the humidity, temperature, air pressure, how nervous you are, arco vs. pizz, etc. John Clayton told me to think of intonation like someone had pounded nails through the tips of your fingers. The points of the nails go down where you think the note should be, but then you can (and should) rock your finger around the point to get the correct pitch.
Anyways, hope that helps.