We know that the consistency of ZFC+"Exists an inaccessible cardinal" implies the consistency of ZF+DC+"All sets are Lebesgue measurable"; and DC proves the existence of non-Borel sets.

J. Truss proved that repeating Solovay's construction by collapsing any limit cardinal to be $\aleph_1$ we obtain a model of ZF+"All sets are Lebesgue measurable", and in that model DC holds if and only if we collapsed an inaccessible. If we collapsed a singular cardinal then the resulting model has the property that all sets are Borel.

It begs the question, if we assume ZF+"All sets are Lebesgue measurable"+"There exists a non-Borel set", can we conclude that there is an inner model with an inaccessible cardinal?

Some clarifications:

1. When I say Borel sets, I mean an element of the $\sigma$-algebra generated by the open sets.
2. When I say Lebesgue sets, I mean an element of the $\sigma$-algebra generated by completing the Borel $\sigma$-algebra with respect to the null ideal.