Old and young, men and women alike, all looked forward to meeting at the local bathhouse.
Literal
Old [also-も] young [also-も], also male-female [object-を] regardless-of, local bathhouse [at-で] together become [nominalizer-こと] [object-を] looked-forward-to.
老いも若きも ('both old and young') is a literary set expression preserved with the classical attributive 若き. ~を問わず ('regardless of') is a formal way to mark a category as not mattering. ~を楽しみにしていた marks past anticipation. 浴場 ('bathhouse') ties into Japan's deep communal bathing culture, where neighborhood public baths historically served as social gathering spots.