There is a possible legal solution, but I AM NOT A LAWYER AND CAN NOT PROVIDE ANY GUARANTEE. Be sure to talk to a lawyer, before you try to do anything.
So the obvious answer is using SciHub and LibGen, but they are not legal. What as far as I know is legal is using peer-to-peer networks, such as torrents, because here the Slovak (and European) legislation should be a few steps (maybe even kilometers) behind the reality. So the solution would be to set up a peer-to-peer network for distributing articles (yes, just like torrents). Every user would join in and offer the access to his articles (authored by him) and could on the other hand get article authored by anyone else in the network for free. This would be just like writing an email to the author, except very automatic.
Since Elsevier, Springer mostly gives the authors right to share their work with colleagues, friends, etc. this would maybe even comply with their policies.
Unfortunately I do not know of such network.
Update: for anyone interested, there are at least promises that the problem will be solved, see this article.