Inkwells and the kind of pens you have to dip in them were quite obsolete even in the 1960s (except for artists and calligraphers, who still use them today.) I'm sure Kirk was intentionally using an archaic reference for the sake of humor.. . . Reminds me of Kirk’s description of childish mischief, including such things as “dipping little girls’ curls in inkwells.” I bet that’s really common with 23rd century kids.
So was "tying cans to the neighbor's trees". You know, blowing in the wind you'd get all that clatter. That's even way before my time.

It was the late 1960's... sexism was still every much alive and tolerated. So, it's no surprise to see some of it littered about Star Trek. Remember, even having a female 1st officer (The Cage/Menagerie) was unheard of then. That's why it wasn't until TNG that we saw female captains (e.g. Garrett, of NCC-1701C) and admirals (e.g. Nechayev for one--there were several who appeared). Anyway, TOS was extremely progressive for the day. It's easy to forget that in the context of today.