First one must assess whether researchers in the area agree that the community should leave Elsevier. If there isn't strong support for leaving Elsevier, then any move is likely to fail, as it will probably involve a new venture (such as a new journal) requiring the support of many, many researchers in the area, perhaps as authors, editors, or readers pressuring their university to subscribe to the new journal. If the research community is not supportive, this may because they have very good reasons or it may be out of ignorance / lack of imagination of the alternatives.
Discussion of the possibilities in your researcher community serves to explore the options, educate, and potentially build support. You may be able to start a discussion of the issues on mailing lists, social media, or run a conference symposium/satellite related to this topic.
If there is community support, there are a few possibilities for actually making the move. In most (all?) cases, Elsevier owns the journal and its name, therefore one cannot simply switch publishers and keep the same journal name. As a work-around, moves have occurred when all or most of the members of editorial boards of Elsevier journals resigned and started a new journal, usually issuing an open letter explaining their action and encouraging the community to submit to and subscribe to the new journal.
A new journal can use the traditional subscription model or be open access.
With a subscription model, one can use a traditional publisher- a non-profit university press may be less objectionable than Elsevier or one of the other mega-profitable corporate publishers. I have started a list of possible publishers. To get started with a new publisher, one must convince them that they will make enough from subscriptions for the new journal to be worth their while. This may be difficult, as new journals are frequently risky. It takes a few years for a journal to receive an impact factor, and may also take years to be indexed by the major databases, and many authors will only submit to journals that have already achieved these things.
An open-access journal can use the author-pays model, in which case a large publisher can provide all the traditional services (manuscript submission software system and reviewing workflow management, layout, copyediting, production, webhosting, accounting, exporting to databases, DOI registration, proper metadata, etc.) or it can be run on a shoestring, with academics handling everything perhaps with a few administrative staff. For research communities willing to submit all their manuscripts in LateX, this is quite feasible but for communities that demand layout (figures and text arranged to fit a standard page appearance and possibly typesetting) be done, this is more labor-intensive.
Several open-source software tools assist in publishing journals. Open Journal Systems is most like a traditional journal publishing platform but I hear it may be difficult for academics to use. Annotum is based on Wordpress and I believe it works by having authors write their manuscript directly in its software, so that it can guarantee that the paper will look exactly as you expect it (WYSIWYG). It is used by PLoS Currents and other journals. All of these tools could probably use more skilled programmmers contributing to the project.
I am only a researcher, not a publisher, so perhaps not everything I have written here is correct. I think we researchers are in particular need of estimates of the person-hours needed to publish and manage a journal by various methods, so that a research community considering a move can budget appropriately / be comfortable knowing what they're getting into.