The answer to this question, in it’s entirety, is yes.  

(I also enjoy me some TOS~  The DVDs are sometimes visible on the shelf behind me in the photos I post of myself doing wacky shit.)

‘Those two characters’ are Julian Bashir, the adorkable human doctor; and Elim Garak, the pansexual Cardassian tailor/gardener/spy/all-around-hypnotically-amazing-guy.  

The man who played Garak, Andrew J. Robinson, decided on the pansexual approach in his first episode, which was to be his only one originally until he proved to be all-around-hypnotically-amazing-guy, and it’s definitely very apparent there. The writers have also commented that that was definitely how they wrote him - as quite attracted to Bashir, but rather content with the knowledge that it was unlikely that Bashir was going to notice (because giant oblivious dork) or reciprocate (because famous for fantastic failures with women despite being gorgeous).  I don’t know anyone who watched that first sequence with them and didn’t think, “Uh, did that grey guy in the watermelon outfit just hit on the doctor?”

This continued for like three seasons in which at least half of Garak’s lines were pretty ridiculous flirting, if not more - and I would say this was to the enjoyment of the actors, who became BBFs to the point of Robinson is Siddig’s son’s godfather - and there’s definitely a spread in one of the 'Romance’ issues of Star Trek magazine devoted to these two.  Garak and Miles O'Brien are literally Doctor Bashir’s most successful interpersonal relationships of any type for a long time and they both start off on some predictably awkward footing, because Bashir.

Eventually, someone decided that what starts looking painfully like a developing and absolutely fascinating interspecies relationship was 'OMG GAY NO NO NO’ (despite that a lesbian relationship was portrayed and not fussed over on an executive level, even though it only lasted one episode) and they pretty much just tore the characters apart and made up random relationships for them to compensate for their lack of each other. (Suddenly Bashir’s obsession with Dax returned, even though he seemed to have gotten over her and was very happy with 'friends’ for seasons upon seasons.  Suddenly the half-Cardassian girl on the station was aged up like ten years to interact with Garak.)  The writers were no longer allowed to give them significant scenes together, the actors were forbidden from speculating about Garak and Bashir’s relationship at conventions. 

I think it stinks a little of 'doth protest too much’ for most of us. 

Possibly also to Andrew Robinson, if his book is anything to consider.  While it isn’t blatantly 'COME TO ME, DOCTOR’ on every page, he was able to write a book in which Garak was explicitly attracted to a few men as well as women, compares Bashir to pretty much every person he’s ever loved, and gives this entire novel-length collection of thoughts and detailed descriptions TO the Doctor as a letter after they’re no longer on the station together.  This from a man who lied to everyone, all the time, about who he was and what he was about.

Nothing would make him happer, he says, than a visit from Bashir.

So while the book is essentially extra-canon, it at least gave Robinson the chance to do what was initially wanted for the character slightly more openly.  And while Bashir is definitely in a relationship by the end of series, his track record with those and his incredible interactions with Garak leave most of us deciding they both move to Garak’s planet and adopt orphans.

So yes. They were so close to being and may have supposed to have been legit, and we are wearing googles.