There's really no substitute for the thud of a real acoustic upright. I've never owned one, but have recorded them a number of times.
I recently got an NS Design CR4M electric upright. It sounds more like an acoustic upright than any 34 scale bass, but it's quite different sounding than a real acoustic. I'm not a good enough player to tell, but my friends who are accomplished on upright tell me that it feels about right, minus all the physical landmarks that they get from the body and neck joint. Of course, it's much easier to move around. One friend plays in a bluegrass band and said it would probably pass, but the NS bass has the most radical styling of the EUBs, which might be a problem.
I have a Turner Ren 5 fretless. The Acousticore strings give it a very distinct sound, but it's not like an upright at all. As someone mentioned above, it is clearly the king of fretless mwah - completely controllable by your touch.
I've also got a Veillette IV fretless 5-string bass. This bass is called the Paris now (and unfortunately got a lot more expensive). This is a really beautifully hand-crafted instrument that is great to play. Mine has Alvarez electronics, but the new ones have the Turner-Duncan DTAR system. This is a 35 scale instrument (same as Turner 5-strings) and came with nylon LaBella strings that are much more organic sounding than steels. It's a very, very nice instrument. The Rob Allens have been mentioned many times here - this is a very similar vibe.
You can get a surprisingly close sound with those LaBella strings on a Godin Acoustibass, which is usually pretty cheap when you can find them. This instrument is build like a Precision with a routed body and thin top and LR Baggs pickup. Again, clearly not a solidbody bass, but no upright.
It's hard to get a real dreadnought-style bass that sounds good even amplified. There was a Martin that was expensive and didn't sound good. The Steve Klein-designed Taylor AB was a marvel of design and construction (this is the one with a wedge-shaped body and the soundhole off to the side), but they didn't really sound good either. I have an old Kramer Ferrington bass from the 80's that actually plays really well and sounds pretty good. I liked the sound of this better than the Tacoma Thunderchief, although the Thunderchief was a very good value.
David Fung