The meaning is that either one shows that one or more of the axioms or some hidden premise is not acceptable for some reason, or one accepts the conclusion; or one can live in a state of "there's something wrong with that, I just can't find it" for as long as forever takes. If that means sleepless nights, so be it.
FWIW, a long time ago I chose to stop talking about particles. I haven't followed the Conway-Specker literature, since I argued some years ago that Bell inequalities cause almost no problem for random fields and no-one has so far contradicted that argument [go at it now, if you like, "Bell inequalities for random fields", J. Phys. A: Math. Gen. 39 (2006) 7441-7455, cond-mat/0403692, if you haven't previously decided it's wrong and decided not to publish a rebuttal]. Conway and Specker introduce assumptions that are fairly reasonable for classical particles but it's easy to exhibit that they do not hold for random fields.
The MIN axiom, in particular, is generally not true for quantum fields, because there are correlations at space-like separation in QFTs, without, because of microcausality, any need for there to be transmission of causal effects between space-like separated regions.
The MIN axiom is also generally not true for random fields. There are correlations, but there is no causality, because there is microcausality at time-like and light-like separation as well as at space-like separation. The correlations, to your taste, can either be pre-existing (the Conspiracy Loophole, if you like, but the conspiracy is probabilistic, not deterministic) or, instrumentally, they're there just because they're there (that is, we observe them, so they're there). The FIN axiom is harder, because then one has to be convinced that negative frequency components of the random field do not cause trouble, but of course classical electromagnetism lives happily with negative frequencies without problems of causality outside the light-cone, negative frequencies occur in QFT loop diagrams without difficulty, and also frequency is not linearly related to energy for classical fields.
For random fields, since they're not well known, you can try my approach in "Equivalence of the Klein-Gordon random field and the complex Klein-Gordon quantum field", EPL 87 (2009) 31002, arXiv:0905.1263 quant-ph, where a few references to other people's work can also be found. It's a niche, of course.
Move on, is what this theorem means; in my case on to random fields, but it's best if we all make different choices. You might, alternatively, be happy to drop locality and/or classical states and observables and/or something else, or to lose sleep.
On the other hand, I think you could do much worse than to accept the answer on Luboš Motl's blog, which I found very interesting, so I voted up his comment on Roy Simpson's Answer as proxy.