Whether King Arthur existed is a question that many historians consider largely either unanswerable or likely in the negative.

The major issues at hand are the scanty sources for Anglo-Saxon and Celtic England prior the later centuries of the first millennium. King Arthur's life, if real, is traditionally dated to the 5th-6th centuries CE, a time period that lacks much documentation, with the only major contemporary source in England at the time being Gildas (who was used as a source by Bede in his Ecclesiastical History of the English People, written in the eighth century CE).

Here are the issues with our sources.

The earliest source to mention King Arthur is the work of pseudo-Nennius called the Historia Brittonum written in the 9th CE. This work has been cited by scholars such as Arthur Breeze, but Breeze is also quite critical of it ("The Historical Arthur and Sixth-Century Scotland" Northern History 52, 2015, accessible here). Unfortunately, this source's reliability has been constantly questioned. There are also a number of contradictions, because the earliest Welsh traditions of King Arthur lack any mention of the Battle of Badon, while the Historia Brittonum does mention it, as Breeze notes. Thus, we have little reason to simply accept the Historia Brittonum at its word. Simultaneously, the Historia Brittonum presents a legendarized and distorted view of England's past in general. It is the second source to claim the presence of Hengest and Horsa in England (I think they were historical, but that is a side note), but does so with a high degree of supernatural and magical events. For example, Vortigern meets his end and dies when St. Germanus calls down fire from heaven and consumes his household as punishment for Vortigern's sins. Likewise, the Welsh sources (mostly the Annales Cambriae) betray evidence of having knowledge of the Historia Brittonum, meaning they are not independent evidence, and likewise, they show evidence of having been compiled over time and the Arthurian elements may have been late (John T. Koch, "The Celtic Lands," in Norris J. Lacy (ed.), Medieval Arthurian Literature: A Guide to Recent Research [New York: Garland, 1996] pp. 239–322).

Additionally, there's a stark absence of Arthur being mentioned in other sources. Unlike Gildas who can at least corroborate the three boats story in Bede, thus supplying a possible basis for some Hengest and Horsa legend (and Hengest and Horsa are vastly more attested in English sources, being found in Beowulf, Bede, the Historia Brittonum, the Anglo-Saxon Chronicle, all of Geoffrey of Monmouth's writing, etc.), there is no attestation of Arthur in Bede, Gildas, or the earliest Anglo-Saxon legends.

So, the question is this... where is the evidence?

Of course, because of paucity of sources for early Britain, the reality of the situation is that King Arthur very well might have existed, but there is nothing that survives about him outside of legend. A very similar issue happened with the Burgundians, who are likewise very poorly attested in ancient sources, but the figures of Gunther, Gibica, Giselher, etc. all proved to be the basis of The Nibelungenlied, the Volsunga saga, and others (see Edward R. Haymes and Susann T. Samples, Heroic Legends of the North [Routledge, 1996]). In this case, however, our certainty arrives from a singular surviving legal text, the Lex Burgundionum, which mentions them. Even Brunhild was a historical queen of the Merovingians (see Theodore M. Andersson, The Legend of Brynhild [London: Cornell University Press, 1980]) who married Sigebert I (the likely basis of Sigurd/Siefried according to several recent analyses, see Catalin Taranu, "Who Was the Original Dragon-slayer of the Nibelung Cycle?" Viator 46, no. 2 [2015]: 23–40).

So, whether King Arthur existed, and whether he was even a king or not, will likely forever remain a mystery. There are no ancient sources that attest to him, but if the Germanic and Celtic tradition of remembering ancient kings for hundreds of years in legendary oral tradition is anything to go by, he may have very likely been a historical person, who has been glossed over by the stories attributed to him, much as Gunther and Brunhild were.

Save for a few exceptions like Arthur Breeze (whose work has been staunchly challenged) most scholars do not think Arthur was historical, or if he was, they do not think any historical figure is recoverable beneath the legends.