As someone who has had the misfortune of writing longer short stories/ novellas, I sympathize with anyone who has tried to publish them because I have spent so much time scouring the internet screaming WHY WONT YOU TAKE MY MASSIVE TOME THAT ISNT QUITE MASSIVE ENOUGH TO BE A NOVEL. Below is a working list of literary magazines, contests, and small presses that publish longer short stories and/or novellas. This is a list for literary fiction, not genre fiction and of course those lines can be blurred but if you've written something that can be considered genre you should probably familiarize yourself with the magazine because these markets are indeed different than the stuff I see in genre magazines. This is not a exhaustive list, it is a working list I will update, and mainly includes magazines I know to be well respected and presses I have seen with my own two eyes. (I have submitted to many of these.)
Magazines and Small Presses
Alaska Quarterly Review, up to 50 pages. One of my white whales! A great magazine that has been beloved for so long. They only take hard copy submissions.
A Public Space doesn't list an upper length limit, and specifically says "novellas and novel excerpts are always welcome."
BatCat Press takes stuff of any length. They publish weird, beautiful books; this is not a standard market--I think they are looking for stuff that is "out there."
Blackbird is a well-respected online magazine. If you have a piece that is 8k or over, you can query first to see if they're interested. (that said, the magazine sometimes is on the higher end of how long it takes to get back to writers.)
Boulevard takes stories up to 8k
Conjunctions is one of the most well-respected magazines out there. It has no official restriction on length, but does not take simultaneous submissions and submissions are hard copy.
The Collagist (a magazine run by Dzanc Books) takes stuff up to 8 k
Online magazine failbetter says they publish novellas, but given that it is online, I would page through to see how long their stories actually run.
Fiction is old-school respectable. Their website says they prefer under 5k, but would consider longer works. (read: long shot)
Gettysburg Review up to 10k. Interpret as you may: "We do not publish genre fiction—mystery, crime, science fiction, fantasy, and the like—but are certainly not opposed to considering work that self-consciously employs the tropes of more formulaic writing for more sophisticated literary ends." They take about half a year to get back to you, but are a great market.
Malahat Review up to 8 k, Canadian lit mag that discourages simultaneous submissions. They also run a novella competition every other year for submissions 10-20k
Massachusetts Review takes stories up to 8 k, but they also have a digital release program called Working Titles that has a 7-25k limit.
Missouri Review "While there are no length restrictions, longer manuscripts (9,000 to 12,000 words) or “flash fiction” manuscripts (2,000 words or less) must be truly exceptional to be published." Extremely competitive market.
One Story 3-8k, this should be one of your top markets. Given that they only publish one story per issue I wish they would take longer stuff :(
Paris Review does not have length restrictions; that said, they take hardcopy only and I don't know anyone who ever got in there off the slush pile.
Ploughshares Solos 7,500-20k, extremely competitive market
New England Review 20k
Nouvella Books 10-40k. They publish beautiful little (literally) books. This isn't run like a standard literary magazine (because it isn't one), but they view their novella line as a way of investing in and launching emerging writers.
Puerto del Sol, up to 10k
Seattle Review is specifically looking for longer works--stories over 40 pages--yay! The bad news is 1) they don't take simultaneous submissions (BOO.. although they did only take 1.5 months to get back to me), and 2, they go out of their way to say they won't take genre. Clearly there are stories that are intended for Clarkesworld but there's also literary writers out there that write weird stuff, and seeing restrictions like that makes you hesitate to submit.
Tin House up to 10k. A top market but I don't know anyone who's ever got in off the slush pile. (for the magazine- for the blog, yes.) You can submit, but it's one of those "submit it and forget it." Seriously. You might not hear back for more than a year so it's not a "they're taking a while so I must be making it up the decision chain!" thing.
Incidentally, and I'm thinking of one magazine in particular, don't submit anywhere that has a $20 or higher submission or reading fee. Magazine submission fees and contests are two different things. Submission fees are controversial, but a few bucks doesn't seem unreasonable to me given how literary magazines are struggling. For more than a few bucks--you do the math, it isn't ethical.
All of these prizes are yearly or close to it and have entry fees. There are of course more contests than just these, but these are the ones I know of that I would call good. (ie, the entry fee/ prize ratio isn't eyebrow-raising, the judges are well known literary writers, the winners go on to do well in their careers). Maybe this is an obvious tip, but if you're submitting to a contest you should wait until a few days before the deadline. When submitting simultaneously, if you submit to a contest in January that doesn't have a deadline until March, there's a chance your piece will get picked up somewhere else first and then you can't get your entry fee back from the contest when you withdraw. Submitting early doesn't help you in any way.
Big Fiction Knickerbocker prize 15-40k
1888 (formerly Black Hill Press) Plaza Literary Prize, 10-40k (you have to dig through their terms and conditions page to find the word count). They publish beautiful books--I love the artwork they use.
Black Lawerence Press small press that publishes full length novels, poetry, collections, anthologies, etc. The Big Moose prize is for "novels" but their submission guidelines say 90 to 1,000 pages--90 pages would put you within the range of a novella.
Miami University Press novella contest 18-40k
Texas Review Press Clay Reynolds Novella contest 20-50k