There was a thread on RPG.Net about running Fate "aspectless", since the poster's group wasn't really getting into using aspects descriptively (i.e., just saying they spend a Fate Point to get a bonus or reroll without a lot of exposition). It doesn't sound like they were using aspects incorrectly, more like not really getting into it. The poster wanted to know if it would break anything to just use the Fate Point economy, with the equivalent of compels, Create Advantage, invokes, etc. but without having aspects. I'm going to expand on my reply to that thread a bit.
Fred Hicks has said that aspects are the least mechanical part of Fate Core. While having too many can bog things down, the number of aspects doesn't affect power level so much (since they're really all the same, mechanically speaking). As such, going completely aspectless in my mind doesn't really hurt anything...but at the same time, I believe that you really can't go completely aspectless. This isn't due to some, "You have to play the game this way!" dogmatism, it's because of my emergent view on aspects. I think that in spirit, when we play any roleplaying game we're using aspects whether we know it or not. When you describe a door as an Iron-Banded Door, you're tapping into the imaginary space that tells us what an Iron-Banded Door is like. Or when you say say a chasm is Too Big to Jump Across. Whether that means having to roll an Overcome with the proper justification to get through it, or defining the door as having 20 Structure Points and 5 Armor that has to be battered down or somehow bypassed - it's the same thing. We use language to communicate the properties of things, and aspects are language. They come about naturally from simply ascribing descriptors to people, places or things.
Obviously in terms of actual aspect use in Fate Core it's a little more than that - mostly because of the negotiation process that can take place with justifying how a 98-Pound Weakling can rip a door off its hinges (protip: they can't, at least directly, and I'd deny that invoke). But the processing, for me, of whether or not that's possible is pretty much the same regardless of whether or not 98-Pound Weakling is in the form of an aspect or some other truth within the game system. And, in the end, if players are just offering, "I invoke my Trained Swordsman aspect for a +2. I roll to hit." in a Fate Core game, they're likely just doing the equivalent in any other roleplaying game (and, honestly, if they're narrating how their blade whistles through the air toward their doomed opponent when playing another game, they're likely to do it in Fate Core too).
So, to that end, I'd say that it's perfectly possible to play Fate "aspectless", but only to the degree that you can get away from attaching descriptors and qualities to things in a type of game that is mostly about describing and qualifying things. Which, unless you're games take place in completely white empty rooms with characters only vaguely taking any kind of action, means never.