Oops, I missed the strictly positive part.
Let f(x) = 1/2 if x is rational and be equal to 2 if x is irrational. Let g(x) = 2 if x is rational and be equal to 1/2 if x is irrational. Both f and g are not Riemann-integrable (I'm assuming this is the type of integrability you're talking about) in [0,1] since they both have an uncountable amount of discontinuities in the interval. But fg = 1, which is integrable.