One way to express that would be:
Omnia possibilia sunt.
All things are possible.
This actually comes from the Vulgate, where Jesus said:
Apud homines hoc inpossibile est,
apud Deum autem omnia possibilia sunt. (Mateus 19:26)
With men this is impossible; with God, however,
all things are possible.
Everything vs. anything:
Concerning your comment about whether anything and everthing can be interchanged, I believe it's common for translators to do so. Technically, it doesn't express the idea in the exact same way. Rather, it's considering the same thing from different pespectives: everthing considers the subject collectively, whereas anything considers it from an individual point of view (i.e. each thing considered in turn). Even so, I believe it's often important for the translator to express thing in a way that seems natural.