Implicitly contradicted by Livia's lorebook profile, though:
Robbed of her parents at a young age, the Garlean girl was taken in by Gaius, who had known her father. Though leaving behind her younger sister Lucia was difficult, Livia found Gaius to be a stern but thoughtful guardian. In time, Livia came to see him as more than an adoptive parent, and joined the army to follow in his footsteps, where she eventually became his right hand.
(Page 185 of EE1)
Given Livia made a name for herself in putting down a rebellion in Dalmasca (the Barheim Incident), she would have been rather high-ranking in the XIVth at the time. And given her age at the time of ARR (and her death) was twenty-six, either she was so talented that she shot up through the ranks like no other (I don't believe Gaius would have committed nepotism, or he'd be a far worse person than my already low opinion of him) between being hypothetically adopted as an adult, or she joined the military at a young age and progressed through the ranks as per usual (if rather quickly).
For comparison, Milisandia of the Ruby Weapon was reportedly 30, four years older than Livia.
As for the "one-sided infatuation", there was the point in the middle of the ARR MSQ where we saw a cutscene of Gaius delivering his New World Order speech, and then tells Livia "my quarters, one hour". Which was really creepy at the time, since it basically meant Gaius was going "we shall usher Eorzea into a new age of order! Also I'm horny."