I've only ever seen two failures, one sucked water and bent a rod (owner's fault) and the other was a fluke, a cam bearing spun out of its bore and it lost oil pressure at 200k. Everything else still looked like new inside. Engine went back together with new cam bearings (old cam) and that was 60k ago or so.
Generally, a 3800 will outlast the car around it.
Forever.
FIL 2000 Monte Carlo SS made it to 317,000 before it spun a rod, he replaced the engine.
Used Mobil One from day 1
make sure the intake gasket isn't leaking and keep it from overheating and it'll run forever
'98 Olds Intrigue 3800 quietly leaked coolant from the water pump, resulting in interaction of air and Dexcool coolant, resulting in brown sludge, resulting in distortion of plastic intake manifold and $2000. in repairs. Temperature gauge never read more than a tick higher than normal. GM refused to cover any of the cost of repairs (until later sued in a class action), resulting in me being quite sour on the notion of ever buying GM again. A basically solid engine encumbered with various parts designed to fail. Oh, and to answer the original question: 70,000 miles.
My Buick fell apart around its 3800. When the transmission failed that was it for me. The only other good thing about that car was the radio (aftermarket).
Got a customer that has 3 W-body GM's. One for his daughter, one for his wife and one is his. The daughters is at 517k miles on the original engine/trans. It did have the head gaskets replaced at 390k. The wife's is the "new one" with 306k miles. His has 414k miles. Oddly enough, the daughter's car, everything but one window still works. His has fallen apart around him.
My 3800 in the Bonneville is going strong at 144K. Main issues they have are the plastic upper intake at egr (non-supercharged), lower intake gaskets and coolant elbows. Keep an eye on those (or just change them out) and it will run until the car falls apart around it.
I sold my Bonnie to a friend with 295K and the motor was still running strong. The trans was original and starting to misbehave though.
Had '98 Z34 Monte Carlo, it leaked oil and used coolant for the last 20k miles I had it, but never let me down and still ran great.
I think it was even on the original spark plugs at 170k miles when I got rid of it.
They don't last very long if the coolant passage into the intake that cools the EGR passage gets plugged up. They burn a hole in the manifold which then applies full manifold vacuum to the cooling system, which unplugs the EGR cooling passage, and sucks all the coolant through the engine...
Mostly they will vibrate and shake their way down the road for a pretty darn long time, though.
Mine (a ~98 Series II) killed the main bearings 2 weeks after I replaced the intake and gaskets with the updated ones, but I'm also the guy who spun a rod bearing on a Toyota 5S-FE full of good oil so hard the engine seized, pulling away from a stop sign.
In reply to Kenny_McCormic:
remind me to never let you near anything I own! 
In reply to Bobzilla:
I also wiped a cam on a FIAT SOHC full of M1 15w50, but that's at least sorta common.
Things I didn't kill: GM 2200 LN2(the base caviler/sunfire/S10 engine, I beat the hell out of this one), Mitsubishi 4G15, Toyota 1ZZ-FE(pending).
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