Joint consistency of continuum real valued measurable and Martin maximum What is known about the consistency strength of
ZFC + the continuum is real valued measurable + Martin's maximum?
 A: Martin's maximum implies the continuum is $\aleph_2$, and therefore it is not real-valued measurable, since real-valued measurable cardinals must be limit cardinals and indeed weakly inaccessible and weakly Mahlo and more. 
So unfortunately, what is known about your theory is that it is inconsistent. 
A: This is a supplement to Joel's answer:
In fact, even very weak forcing axioms are incompatible with the continuum being real-valued measurable.  For example, if the continuum $\mathfrak{c}$ is real-valued measurable, then $\mathfrak{b}<\mathfrak{c}$, where $\mathfrak{b}$ is the minimum cardinality of an unbounded family in $(\vphantom{b}^\omega\omega, <^*)$, and $f<^*g$ means $f(n)<g(n)$ for all sufficiently large $n$.
Martin's Axiom (and much weaker forcing axioms) implies $\mathfrak{b}=\mathfrak{c}$, and hence implies that $\mathfrak{c}$ is not real-valued measurable.
Lemma 27.9 on page 304 Jech's Set Theory gives a slightly more general result, along with a proof.  (One needs to note that if $\mathfrak{b}=\mathfrak{c}$, there is a $\mathfrak{c}$-scale.)
