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There is a hg plugin which allows lossless communication between git and hg [1]. So in a way github already exists for mercurial, and since it already exists there is no question as to whether or not it is possible to do.

1: http://hg-git.github.com/

Mercurial has http://bitbucket.org/ -- not nearly as popular but a tad less cluttered IMHO.

Github actually offers support for svn, in a way - see http://help.github.com/svn-importing/ for a bit more info.

It is also possible to use git on top of svn - see http://www.kernel.org/pub/software/scm/git/docs/git-svn.html

I've been experimenting with bzr-git, pushing my bazaar branches to github... so far, so good (though I've not really done any 'social' stuff yet or been branching other peoples work).

CodePlex (most commonly used for .NET projects) has support for Mercurial but I don't think its social features are quite as strong as Github's.