Without further context, "I hear the woman" is the only thing an Italian would think.
"Io sento l'odore della donna" would be more common to say "I smell the woman"
As for what Elena18 says... It could be, but without context nobody would ever think at it. Another elegant way to say "I feel the woman" is "Io percepisco la donna"... but it's not something you will hear often.
"sense" works for this sentence also, but this would be creepy if I "feel" the woman. If it were mutual then the verb "sentirse" would be used which is reflexive and then there would be two-way feeling going on, as in "feel each other".
ocf781- I was marked wrong in the past too, the reason being 'donna' is 'woman' whereas for DL 'lady' would be 'signora'. I disagree w/ DL to the extent that in English at least the two words 'woman' and 'lady' are often used interchangeably as synonyms without regard to social status or etiquette. Still, that's why your answer was marked incorrect.