I'm in the process of planning a story right now, without the intention of sharing it it with others, but I would like it to be comprehensible if I do. It's set in the 1880s and is formatted like a journal which belongs to a doctor, and there would be some words used that people now might not be familiar with. Should I replace these with their modern counterparts, or is it okay to use the period-correct version? If the latter, would it interesting and useful to include annotations (say, as if someone had later found the journal)?
This really comes down to style and the backstory for your journal. If it's supposed to have sat unread in a drawer for 140 years, nobody would have been able to annotate it; if it's been an object of scholarly study throughout that time, there'll be endless rewrites and examinations of it.
For myself, if I were reading a piece of fiction that was set in a certain time and written by a person of that time, unless it would be totally unreadable without translation (say, written in Old English or similar), I would prefer to see the unusual words exactly as they were written and look them up myself if I'm confused. Adding annotations to clarify feels rather cheap to me, and feels as though the audience is being patronised.
It also severely dates the work, because if somebody's reading it in twenty years and finds that some old words have been annotated but others haven't, it feels less like it was written 140 years ago, more like it was written twenty years ago. For example, if I was reading a book which noted that "Gay" simply meant 'happy' and not 'homosexual', I wouldn't be thinking "Ah, thanks for sharing that", I'd be thinking "Oh, this was annotated ten years ago before 'gay' became a general expression of dislike".
If you are after verity, you have far more to worry about than vocabulary. In the nineteenth century the whole style of writing was different. Paragraphs and sentences were much longer than we typically use today and the overall tone was far more formal. Language usage was far more of a marker of class than it is today. (Many of our grammatical shibboleths that are slowly dying out stem from the desire in that period to clearly delineate the speech and writing of the upper classes from that of the lower.)
But stories are always written for a modern audience. Setting a story in the past (or in space, or in wonderland) is simply a device for creating the right stage on which to tell your story. Even those in the historical fiction community who scream blue murder about the wrong number of buttons on a lady's glove don't bat an eyelash as monstrously anachronous speech, opinions, or actions. Almost no historical novelist, for instance, make any attempt to portray the religious or class attitudes that would be common to ordinary people of a given period, except as marks of villainy.
So, your doctor's journal is a device for telling a story to a modern audience, and it only needs as much faithfulness to the period as is required to tell the story you want to tell. Too much will do as much harm to verisimilitude as too little, by distracting the reader from the story. But exactly how much you need to do really depends on the story you are trying to tell.
I sometimes use archaic words but tell the reader the meaning in the same way a newspaper defines words the lower vocabulary reader might not understand.
For example. "Did you do the task I set eow?" "Yes, Lady. It is done." or: The percolator coffee maker sat on the stove.
You can tell that eow is 'you' just by the way it is written. I do caution you in over-used of dialect or different language. If you can't read it aloud seamlessly, then it isn't worth having it in the story. Percolator and coffee maker are almost the same thing.