I'm trying to place a short story that is surreal. So while it has science fiction elements, it isn't really science fiction. At the same time it's a bit too weird to be literary fiction. Does anyone know of any journals that publish this kind of thing? The closest comparison I can come up with for the style of writing is Kafka, or J.G. Ballard.
Fantasy and science-fiction is a genre that's bigger on the inside, and you'll find a lot of weird fiction enjoyed and celebrated within the genre. Surrealism and oddness definitely have their place -- see luminaries such as Kelly Link and Karen Joy Fowler, who have won Nebulas for weird short fiction that doesn't fall neatly into genre boundaries.
The Magazine of Fantasy and Science Fiction is probably the best known magazine that loves weird fiction. They're one of the major heavyweights in print, although they're somewhat less prominent now that digital is king.
Lady Churchill's Rosebud Wristlet is probably most on-target for your request (and has Kelly Link as one of its two editors). It's very niche, but publishes great work, and its focus is precisely weird, surreal, poetic works like you seem to be describing. Lackington's is another magazine that's does literary fiction within the genre.
It's worth reading a couple of issues of each of these, getting a feel for them and figuring out what their style is. In some of them, surrealism isn't a constant feature, so you'll want to hunt for specific stories. To be honest, these stories can pop up basically anywhere (e.g. F&SF, The American Reader, Tor.com...).
It's also worth checking out particular authors who write styles you feel are compatible with your own. Check out all the magazines who have published Karen Joy Fowler -- or, the anthology she edited of 2016 selections. Try Carmen Maria Machado; Sofia Samatar; Catherynne M. Valente; see the range of their stories, and where each one has been published.
It's a big field; I hope you find your place in it :D