We have a language $L \in PSPACE$. Is it possible to reduce it with $f \in LOGSPACE$ to $L'$ and $L' \in DSPACE(O(1))$? Why/why not?
The only languages in $PSPACE$ that can be reduced by the above constraint are languages in $LOG-SPACE$.
Assuming it's a Karp reduction:
A language in $DSPACE(O(1))$ is in $DTIME(O(n))$ (the number of configurations is linear). [Thanks to Yuval's comment]
Clearly, everything you can solve in $DSPACE(O(1))$ you can solve in $DSPACE(O(logn))$; So, you might as well leave a bit flag indicating whether to accept the string or not when you reduce the language by solving it in the transducer directly.
Thus, the reduction function will decide the language and since you are constrained to $LOGSPACE$ reduction you can decide only languages in $LOGSPACE$.