http://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem is a decent survey. In general in the discussion of "status" of the Hilbert problems, there are at least two recognisable routes.
Route A is the more natural for contemporary mathematicians. Roughly speaking it equates with asking first for the version of Hilbert Problem N that has entered mathematical folklore (the tea-room version if you like, "Hilbert N asked if [add translation into contemporary jargon]"), and then giving the update on that $folklore$ version.
Route B involves reading what Hilbert actually wrote in German, comparing with accepted English translations, discussing ambiguities and parsing out the issues where Hilbert deliberately made open-ended remarks. In other words Route B treats the problem set as a historical document, and allows for a degree of quibbling and/or interpretative queries.
The reason these routes don't always give the same answer should be relatively obvious once they are formulated this way. But it is worth making the further point, given the tone of various MO discussions, that 1900 is quite close to the cusp at which "discursive mathematics" gives way to "formal axiomatic mathematics". Also (no one express shock, please) Hilbert did not have a definition of topological space, let alone Lie group. The issues here cannot be resolved by saying that without definitions he had no right to pose problems!
There are quite a number of the problems where the accepted Route A box-ticking answer has been queried. Some of these are worth further questions on MO.