The factor in believability is not setting or genre, but the ability of the writer.
For example, I don't usually watch movies or tv series dealing with love relationships set in the present time because the depicted behavior almost always seems completely unrealistic to me. No one I know treats their family, friends and co-workers like the characters do in Sex in the City or the latest Barbara Cartland novel. Yes, there is some similarity in detail, but in total, the characters are completely unbelievable to me.
On the other hand, I always find the fictional worlds depicted in fantasy and science fiction totally believable. Sure, nothing in them is real, but because they don't set out to claim to be true, I can easily suspend disbelief and indulge in the what-if.
There are contemporary novels and movies that approach topics like relationships in the same, what-if way: what if we were completely honest; what if we lived open relationships; what if I only pretended to be in love; and so on. Novels that are thought experiments (and on that interpersonal level do what science fiction does on the social or technological level) are easy to "believe", because they clearly say that they are not true, and this honesty makes it easy for me to ignore those unrealistic parts that are irrelevant to the thought experiment.
There is, of course, non-genre, literary, realist fiction, both in movies and novels, that is believable, just as there are fantasy or sf novels that are totally believable and they feel real except for the fact that you know they cannot be. There are also, of course, novels depicting real persons and events that feel completely untrue and made up. Not because they are untrue, but because they are badly written.
If you are unable to convey the present day real world in your writing, it is only the statement that it is set in the present day that makes it believable to the reader, but not the writing itself. If, on the other hand, you are able to convey a made up fantasy world, it is only the statement that it is fantasy that makes in untrue, while at the same time it will feel totally believable to the reader.
So you need to differentiate between facuality and believability. The first can only apply to existing settings, characters and/or events. The second can be a characteristic of any writing.
To me the Lord of the Rings does not feel realistic, despite its wealth of detail, because the narrative style is not realistic but that of sagas, fairy tales and childrens books, using formulaic narration, stereotiypical characterization and an ironic view on events that comes from the writings that Tolkien loved and intenionally emulated.
The Lord of the Rings is a legend (of a fictional world), and not a realistic novel. There are realist fantasy novels that emulate the style of autobiographies (Gene Wolfe, Book of the New Sun), historic novels (Geoge R. R. Martin, The Song of Ice and Fire) or adventure tales (Robin Hobb, Farseer Trilogy). If you subtract the ficional world from these books, they no longer differ from fiction set in the real world.
Finally, it is not the detail that causes "realism" or believability. If I write, without any detail at all, that:
then that is believable. But if I write, with much detail, that
Joan wailed very loudly and a lot of tears streamed over her face falling onto her legs and wetting her pants.
then that sounds somehow unrealistic because the writing is bad and the details are irrelevant. If on the other hand I write
Joan cried, her warm tears falling on my hands, and I smiled.
then the added detail adds intensity (and a perverse riddle) that make what I write believable, and if I write
Joan cried in rhythmic hiccups that reminded me of the stuttering of an automobile engine running on its last drops of gas.
then that sounds realistic again, because I (hopefully) found an engaging simile and wrote it well.