We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each n-tuple x = (x_1,...,x_n)∈X^n, where X is an Archimedean vector lattice and n∈N:
(i) there is a vector lattice homomorphism Φ_x: H_n→X such that Φ_x(π_i) = x_i for each i∈{1,...,n}, where H_n denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on R^n and π_i: R^n→R is the i'th coordinate projection;
(ii) there is a positive element e∈X such that e≤max{|x_1|,...,|x_n|} and the norm
||x||_e = inf{λ∈[0,∞) : |x|≤λe}, defined for each x in the order ideal I_e of X generated by e, is complete when restricted to the closed sublattice of I_e generated by x_1,...,x_n.

Moreover, we show that a vector space which admits a `sufficiently strong' H_n-function calculus for each n∈N is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function
calculus, while others do not.