I read somewhere that only 2/3 of the V2 rockets reached their target - but how good was their precision when they reached the target? ±100 m? ±1 km? ±10 km?
From Astronautix: "Tests of prototype V-2's in 1943 indicated a 4.5 km CEP (circular error probable - the radius within which 50% of the shots impact). 100% of the shots fell within 18 km of the target. A radio beam guidance update system was introduced in December 1944, which in tests produced a 2 km CEP. In reality, in the campaign against Britain, 518 rockets were recorded as falling in the Greater London Air Defence Zone of 1225 fired, implying an average CEP of 12 km."
Note that the "implied CEP" of 12km includes the effects of misinformation about where the rockets were landing (news of missiles hitting short of London was suppressed while news of missiles going long was reported, or fed to German intelligence via turned spies) as well as missiles which failed outright due to non-guidance problems, so the actual accuracy of the guidance system in the field was likely closer to the 2km-4.5km figures.
Adding to Russell's answer:
A large hindrance to greater precision of the A-4 (technical name) was the lack of calibration data. At this late point in the war, the German intelligence network in the UK was compromised, giving the British secret service the ability to inject false information on the impact sites of the rockets. This lead the Germans to assume their rockets overshot London, so they re-targeted them to what was in fact the loosely populated area of Kent.
Additionally, many of the missiles burst on the last few hundred meters and failed to detonate, because of the static air pressure. When von Braun's team conducted further tests at the elevated (1300m) plateau of White Sands, New Mexico, no such failures occured.
Source: "Peenemunde to Canaveral" by Dieter Huzel