would the Roman Empire have lasted had they built a wall?

 No the empire wouldnt have survived, even the mightiest walls cant keep an enemy out forever, look at the walls of Constantinople, it may have reduced the pressure on the frontiers but the wave of migrating people has to go somewhere.

 They pushed on Romes frontiers for centuries a big wall is just going to slow it down if your lucky, the pressure would have built and built until the frontier spilt over and the barabrians overran it anyway.

 Romes weakness was never its frontiers but its ability to adequatly defend them which was fine except for its major achilles heel, the tendency to fall into civil war practically everytime an emperor died in some periods. Civil war means the recall of the legions to other parts of the empire to fight other legions who should be guarding the frontiers, which leads to invasion which leads to a crisis etc etc.

 A frontier be it a river or wall is useless if its not defended well enough, and quite simply Rome could not hold its frontiers in check so a wall would have made little long lasting difference.

 Not to mention the logistics of building such a monumental wall would be astronomical.

I would say this hypothetical wall, probably along the Rhine would have been doomed.

As Frederick the Great said "he who defends all, defends nothing".  The burden of taxation and the rampant inflation in the West, along with its accompanying destruction of the urban middle class in order to support the huge Roman army had stretched the Empire past the breaking point.  The resources to build such a wall would have were simply not there and even with half a million soldiers the Roman Army was still not in any position to be reduced to spare resources for such a project.

The frontier troops at the time were almost worthless militarily and would have fled before any major sustained effort by the Barbarians to breach the wall, and the good troops that did remain were required as a mobile reserve, not fortification troops.

I think what the Romans perhaps should have considered would be to make an elite light cavalry unit to raid barbarian camps and wreak havoc within Germany itself.  Such a form of raiding/terrorism might have kept the Barbarians off balance enough to mitigate or perhaps stop some Barbarian tribes' attacks on the Empire.

However, after at least 300 a.d., the fate of the empire in the west was a forgone conclusion unless there were massive structural changes within the empire's economy, military, and government.  All the walls and cavalry in the world couldn't stop that.

As has already been said, the resources and manpower required to build such a wall would be impossible in certain regions of the Empire.  In the late Roman period, since the army was composed of largely non-Roman mercenaries and barbarian federates, defending the fortification would have been difficult.  Even if they could defend it physically, the chances of betrayal or sabotage from within would be too great when the army had so many non-Romans in it -- the "Roman" guards could just decide to open the gates and let their fellow barbarians in!

Look at Britain, it obviously didn't work there (nice try Hadrian by the way ). For one thing the construction of a wall in the East to protect against the Persians is an absolute impossibility considering the topography and what it would have taken to actually construct it.

As for the Rhine and Danube frontiers a wall again may have been beyond the capability of construction. We forget China had a MASSIVE amount of human labour resources available for the task of its wall, population density in the Roman Empire was far less and the task would have taken up enormous resources and most likely centuries to actually complete. Next the wall needs to be maintained and defended, and again from what we know of the actual number of troops Rome was able to maintain the manpower to defend the walls wherever trouble might abound along those thousands of northern kilometers was beyond their capability. The barbarians were only increasing in their technological sophistication and would have developed the siege equipment necessary to overcome such an obsticle. Besides that we ignore that Rome was decaying internally, how long before some hard pressed pretender Imperator decides to open his province's defences and invite barbarian armies to help him wage his civil war? And how long before they decide their host is in the way of their own designs........

To sum up, a wall is something that can't be feasibly constructed or maintained so no I don't think a wall would keep out Rome's enemies.