So you cannot have a story without a protagonist any more than you can have one without an antagonist. What you seem to be asking is "can you have a story with no heroes and only villains?"
Short answer is yes, but it's important to remember that "Hero" doesn't mean "protagonist" and villain does not mean "antagonist".
To explain, the protagonist is always the main character, and traditionally the hero of the story, but not always. They are the character through which the audience of the story experiences the story and watches their journey as they try to accomplish their goals. When they do, the story ends and the protagonist lives "Happily Ever After" (except many don't but more on this in a bit). The protagonist is always human or a personification (something not human, but given human qualities that the audience can identify with and understand because as of time of writing, humans tend to be the most common target audience of all known writers.).
Antagonists is never a main character... or even a human or personification. Antagonists are elements of the story who directly oppose the goals of the protagonist. While their opposition to the protagonist might not be personal from their point of view, for the protagonist, it is personal and they must react and over come the antagonist's actions to achieve their goals. Again, the antagonist need not be capable of human thought, malice, or even external to the protagonist. They are always defeated, if not outright and total, at least from the protagonist's point of view (and thus the audience view as well).
Now, to contrast with heroes and villains, this is less about role in the story and more about the morality of protagonist and antagonist as dictated by the writer and hopefully agreed upon by the audience (though sometimes the audience morality is challenged by a character acting against societal morality norms.). A hero is considered morally good, while a villain is morally evil. And the oft popular middle grounds (anti-hero and anti-villain) are types of heroes and villains that can be best summed up as "heroes who are right for the wrong reasons" and "villains who are wrong for the right reasons." To give some film examples (from the same film no less), the members of the Guardians of the Galaxy have traits of Anti-heroes (Sure they saved the galaxy, but they're motivated by their own self-interests (Quill, Gamora, Drax) or general sociopathy (Rocket, who got on board because he could blow things up) or out of loyalty to terrible people (Groot is only involved out of respect for the other four). Similarly, Thanos' villainy is motivated by a sense of altruistic goals (Problem: People are suffering because their worlds can't sustain their population. Solution: Kill half the population so the other half may live. Problem: That's horrible and how do you determine who lives and dies. Solution: Create a method that will take any biased out of the equation. Everyone has a 50/50 chance of surviving. Problem: Whoever does that will be blamed by trillions of people. No one would do it. Solution: Fine, I'll do it myself.). In fact, he resists the temptation of using his newly acquired god-like powers to hurt people unless they directly try to stop his goals... and even then he still regrets that he has to and empathizes with the heroes who try to stop him.
With that in mind, it's also possible to have a villainous protagonist (very rare. Most are actually anti-villains to make them sympathetic) and a heroic antagonist. Consider the characters in Breaking Bad. Walter White is clearly our protagonist but must resort to villainous actions to achieve his goals (initially afford his cancer treatment... but when he achieves it, we learn that affording health care isn't enough and he wants more money for security, comfort, and power. And when he realized he had that, he then wanted to get away without consequences.). His story starts off as anti-heroic or anti-villainous and he seems like he is doing bad things for good reasons (if I make/sell drugs, I can protect my family) but when given chances to get away, it becomes apparent that it's not about other people, but his desire for respect, status, and power that he never had in life. His famous "I am the one who knocks" speech is all about the fact that people perceive him as powerless, when in fact, he's really powerful and ought to be feared. This is carefully crafted and can be seen throughout the show.
In a season one episode, Walter is invited to the birthday party an old business partner who Walt convinced to buy out Walter's interest in their company because Walt believed the company's worth had peaked... only to later realize the company still had bigger profits to make... rewarding the partner's riskier investment over Walt's prudent investment. Walter's jealousy of his old friend is on display and symbolized by his very small gift that he brags his old friend would love is placed beside very large wrapped boxes and makes Walt feel inadequate. When it comes time to open the gifts, we learn that many of them are quite lavish, further concerning Walt. When Walt's gift is opened, and the crowd of wealthy party goers falls silent as the friend reveals the content to be a lone package of Ramen Noodles, $0.99 at a convenient store. Except, as the friend explains, it's not that at all. He recounts his and Walter's college days where they were so broke this package was a luxury meal and not only that, the company that sold them went under years ago and you can't find them anymore. He appreciates it for what it is, and while not saying so out loud, it's clear he felt Walt put more thought and effort into his gift than the rich friends. After all, anyone can by expensive flashy gifts, but it takes a close friend to think of finding a cheap guilty pleasure that is no longer available on the market.
Of course, by this point, Walt is so consumed in jealousy and feelings of inadequacy, he takes the genuine appreciation of his friend as an attempt to spare Walt from embarrassment and later accuses his friend of crocodile tears. Whether Walt's perception is right or wrong, the fallout between the two is entirely Walter's own fault for being to proud to accept the help of his friend, whether it's out of earnest respect or pity, and it's important foreshadowing as to the cause of Walt's problem's later. Here, we see the first and most persistent of Walt's antagonists: Walter's Pride. By this point in the series, it's clear that Walt's lot in life is entirely his own fault and nobody else's. He made a poor choice in hindsight, but it was probably safer back when he chose it. Yet he blames his friend for his own position and refuses his friend's help (genuine or not) to prove to his friend he still made the right choice. Throughout his life and in the series, every misfortune he suffers is one of his own short sightedness and making and yet he constantly blames everyone accept himself for his problems.
But he also has two antagonists. The first, Gus Fringe, is the traditional villain antagonist and his antagonism to Walts goals are easy to understand... because they are the same goals as Walt. Gus is practically a clone of Walter White and while they are both respectful of each other and their talents and skills they bring to the table, they both understand that their goals will lead them to a conflict because they both have the same goal to become the top Meth Dealer in the area... and for one to be at the top, the other has to fall.
But Walt's other greatest foe is his brother in law, Hank Schrader, who's goal is to make a successful career in the Drug Enforcement Agency (DEA), which requires him to bust large drug operations. Given that Walter's goal is to become a drug kingpin, this would naturally put them at odds. Hank's obliviousness to Walter's activity is what gives Walter the edge but when Hank actually learns of Walter's role in his big case, it becomes a contest of trying to thwart each other, and Hank soon finds that he under estimated the lengths that Walt would go to to protect his drug empire.
In a straight story with a hero as the protagonist and a villain as the antagonist, Hank would be the protagonist while Gus and Walt would be escalating levels of villainous antagonism. But because Walt, the villain, is the focal point of the story, it falls on the hero to be the antagonist. Hank isn't a bad guy. Sure, his sense of humor is a little blue. But he's heart is in the right place and he is clearly shown to be a very moral person. But he isn't the protagonist. The show never features him on the DVD box art. It's all about Walter White.
And this isn't the only work that does this. The aformentioned Avengers: Infinity War gives way more screen time to Thanos, the a mo-cap CGI villain, than it does to all it's heroes, who could easily offset their actor's high costs by the fact that all of them require significantly less CGI to show on screen... yet the Mad Titan's journey is the one the audience is told they should be following in the film, from beginning to his victorious end.
Wreck-It Raplp provides Meta example in that it's premise is that many classic video game villains are just doing a job and are actually decent people and often friendly with their heroic counterparts and each other. In fact, they even have a support group for each other to help them with their difficult role as bad guys (Bad-Anon) and they admit they've been trying to get Ralph to come to to meetings for years, possibly hinting that they were well aware that his dislike with his job was causing his depression for quite a long time (while his hero counterpart, while friendly, isn't able to see this despite working with him continuously for 30 years.). As the film wears on, the NPCs of Ralph's home game, who have been dismissive of Ralph's role in their life, are forced to admit that a video game without it's bad guy is just as broken and in danger of being unplugged as a video game without it's hero. In fact, they are just as, if not more important than the good guys... they move the plot along (when Ralph doesn't show up, they have to beg hero Felix to stop waiting for his cue from Ralph... which demonstrates the reason a antagonists exists... because protagonists are only capable of reacting to problems. They can't create problems that they can then solve... but Antagonist can.). In fact, the villain of the film is explicitly stated to be a heroic video game character, that couldn't stand not being a popular game and ended up killing another game hero in an attempt to keep himself in the lime light and this tends to be a problem with many heroes... they tend to buy into their own hype despite the fact that the are dependent on the bad guys to achieve success. Felix relish his life of the party status at the 30th anniversary game... and while he's not antagonistic to Ralph, he does have to come to grips with the fact that his own ego played a part in Ralph's departure from their game.
TL;DR: Your Protagonist can be bad, and that's good. They can never be good, and that's not bad. But without them exclusively, there is no story.