Peer review is done for free by the community in any case, so this would not be a major problem. Most suggestions for a post-journal system I have seen so far are about setting up electronic journals with editorial boards that send out submissions for peer review as usual.
Moving away from journals seems to be largerly an issue of missing consensus and inertia. Also, where you publish is important for hiring committees, existing journals have a lot of credibility, newly sprung up ones not so much.
We already have model for future journals: http://www.emis.de/journals/SIGMA/
SIGMA is just an overlay over arXiv with added peer review. Also it has non-zero impact factor.
Moving away from journals seems to be largerly an issue of missing consensus and inertia. Also, where you publish is important for hiring committees, existing journals have a lot of credibility, newly sprung up ones not so much.