Sole functions on three dimensional projective space? Let the base field $k$ be algebraically closed. As stated in my question title, are the sole (algebraic) functions on the algebraic variety of $3$-dimensional projective space (over $k$) the constant functions?

If you by "algebraic functions" mean morphisms $\mathbb P^3 \to k$ that are regular everywhere (=no zeroes), then the answer is "yes". This is true for example because $H^0(\mathbb P^3, \mathscr O)=k$: the only global sections of the sheaf of regular functions are the constants. See for example Hartshorne, Theorem 5.1 in Chapter III.
If you by "algebraic functions" include rational functions, that is, functions that are locally the quotient of regular functions, then "no", $\mathbb P^3$ have many of these. 
