Rob Allen basses are perhaps one of the best sounding acoustic-esq bass guitars, but they ain't cheap but do sound lush - jump to 2:15 to hear the bass solo:
The cheaper alternative, and often referred to as the poor man's Rob Allen, is a Godin bass. Basically though, any electric bass will never sound exactly like an upright bass due to the differences in physical shape etc. But, if you get a fretless bass, some nice flatwounds, turn all the mids off your amp and play over the neck you can get somewhere close. The Ashbory system is good as well, but it's so short scale that intonation with such thick strings that break often is a bit of a turn off for me.