How is the triferric tetroxide of iron possible?
Iron, like many of the Transition Metals does have a variable valency, it can valencies of 2+ (ferrous iron), 3+ (ferric iron) and in some cases, 4+ (tetravalent iron).

In regards to $\ce{Fe3O4}$, according to the Wikipedia page Iron(II,III) oxide,


  contains both $\ce{Fe^2+}$ and $\ce{Fe^3+}$ ions


A 2-dimensional diagram of $\ce{Fe3O4}$ is below, from the relevant ChemSpider page:



A 3-dimensional model is below (from Wikipedia):



red indicate oxygen