Lower bound on space of DFS keeping the running time linear
The questions as stated seems to be beyond reach. Without the time constraint, a superlogarithmic space lower bound on your question would imply that $\mathsf{L} \neq \mathsf{NL}$, a famous conjecture in complexity theory.

What you are looking for is known as a time-space tradeoff, and these are not known for the RAM model, though similar results are known for restricted models such as the cell-probe model and comparison-based algorithms with memory (modeled as branching programs). 

Time-space tradeoffs for reachability have been studied under the JAG model and its variants, which is a pebbling-based model. The best known time-space tradeoff, due to Edmonds and Poon, implies that polynomial time requires space roughly $\Omega(n/2^{O(\sqrt{\log n}})$, and this has been achieved by Barnes et al. in their paper A sublinear space, polynomial time algorithm for directed $s$-$t$ reachability. (The work of Edmonds and Poon is cited by Barnes et al.)