Well, you said you wanted an honest review, so please don't take what I'm about to say personally and above all, don't be discouraged. Most of us here started out modestly. I'm still somewhat embarassed by my first fan fic story.
While not a particularly original story, the Captain falling in love with a subordinate and the consequences this might lead to can make for interesting storytelling. Certainly, there's a lot of dramatic potential in your story. But, unfortunately, the way it is written prevents it from reaching that potential.
You try to squeeze info dumps about people into parts of the story and into dialogue where they seriously hamper the flow of the story. The worst example in my opinion is this (emphasis mine):
Jim, who was lying on a blanket that he had brought along with a picnic basket, answered Anna’s question. “Well you started to work on the Enterprise in Sick bay two years ago as a nurse. It was just after your husband, and twin daughters died due to a Romulan attack at the last Starbase that you worked on. It took me six months to be able to ask you out, so we have been seeing each other like this for a year and a half.”
So, Kirk and Anna are having a romantic picnic at the beach and he chooses that moment to remind her of a huge personal tragedy? That comes across as very unsensitive. I don't think anyone would talk that way. And why does she need reminding about the length of their relationship anyway? If I had been Kirk in that situation I would have seen it an a rhetorical question.
I understand that you want to tell us all about your characters but it makes for much better reading if you work those infos into the story in a more organic way. If you wanted to inform us about Anna's tragic background at this point (where it wasn't really necessary, but that's your choice) it could have been done by them talking about it or Anna thinking about it, for example.
Another example of this is this passage:
After that she pulled away a little and looked in his eyes and said, “Look at it this way, you will have four days to decide where we are going in this relationship. You can think about this while you are away on the scientific journey that you and the crew are going on, while I am at the conference which Dr. McCoy told me to attend.
The most important thing about dialogue is to make it sound authentic. I don't believe for a minute that anyone would speak like that. Anna would assume that Kirk knew about the conference and the scientific journey (what is that, anyway?) and would use shorthand for it like this maybe: "You can think about this while I'm away at the conference." Then, in a later passage, e.g. when Anna goes to the conference you can add in info about it and why she's there.
Another thing I found weird about the dialogues in the story was that Anna called Kirk Captain and he called her Miss Anna a couple of times. But maybe that was intended to be playful?
There are some very tense or dramatic scenes in your story but you don't make them sound dramatic at all. Prime example:
When all of a sudden there was a massive explosion and then lots of smoke filled the room. Then Anna saw them, Klingons, and they were here on Starbase 17. She thought they must have transported here and then used some kind of weapon to blow a hole through the wall.
I would suggest adding in more detail and more atmosphere, so to speak. There's a massive explosion - How does Anna react? How do the people around her react? What might they say to each other? Wouldn't they try to take some protective measures, especially with the children? How would they react if they suddenly saw a Klingon in front of them?
I suppose all these things can be subsumed under the general rule of taking your time with the narrative. The readers don't need to know everything about your characters right from the start. In fact, it's more fun to learn more about them as the story unfolds. Try to describe what's happening and possibly also what they're thinking in some detail. Otherwise, it won't be very interesting or gripping to the readers.
There are also some things in the plot that are unclear, e.g. how did Krell figure out that Anne was important to Kirk or was that coincidence? And how did McCoy find out about the Klingons dying?
Well, I hope I didn't come off as harshly or anything. I think if you work on the issues I've mentioned you can improve your writing greatly and I hope you keep trying.
As for the question on the number of chapters, the length of the story is really your call as an author. It should be as long as you think it needs to be to tell your story.