The square mile around the Colosseum in Rome seems a likely candidate for most deaths using conventional weapons. Within a mile of the Colosseum were the Circus Maximus and the Roman Forum where there were also thousands of deaths. As most of these deaths occurred during the Roman Republic and Empire, the most commonly used weapons would have been knives/daggers, swords and spears, but many were also strangled or killed by animals.
The Roman Forum was where thousands of captives, criminals and even Roman soldiers were executed, especially before the Colosseum was built. In 206 BC, for example, 4000 or 4500 soldiers of a rebellious legion were executed there. Nearby were the Mamertine Prison where prisoners were either strangled or left to starve to death, and the Tarpeian Rock where people were executed by being thrown off the edge (though I'm not sure if the latter is within a square mile of the other locations).
The Circus Maximus, dating back to at least the 6th century BC was also the scene of thousands of deaths and executions, especially before the Colosseum became the main arena for gladiator fights. Julius Caesar upped the scale of gladiator fights long before the Colosseum was built and, according to Pliny the Elder, Caligula (died 41 AD) had 20,000 gladiators (but perhaps we should make this 10,000 as Pliny was known to exaggerate). It was most famous for the thousands of chariot races held there over the centuries until the 6th century AD: one chariot racer, Gaius Appuleius Diocles, took part in 4,257 races over 24 years, but he was an exception as most racers had very short careers due the high death rate.
For the Colosseum, coming up with a definite number is impossible, but we can at least begin to make an estimate based on the pieces of evidence we have. For gladiators, Keith Hopkins and Mary Beard in The Colosseum estimate 2,000 per year (with big fluctuations) for Rome and base this on one in six gladiators dying in combat in the arena. The estimate of one in six is in turn based on evidence from Pompeii; as there is also evidence from Campania that sometimes half were killed, one in six seems fairly conservative. This gives 710,000 gladiator deaths in Rome over the 355 year period the Colosseum was used for such fights. Many of these deaths would have been criminals forced to fight as gladiators (often in large groups) rather than highly trained professionals going one on one. As the Colosseum was by some distance the biggest, most important and most used venue in Rome, it is highly likely that most of these deaths occurred there (Wiki lists only two other arenas in Rome during the time of the Colosseum, one of which was for training).
These deaths occurred both in major events and ones which have not been recorded in any detail by history. For an example of the former,
the Emperor Trajan, to celebrate his conquest of Dacia (roughly modern
Roumania), gave games in AD 108-9 lasting 123 days in which 9,138
To the above we must add executions and spectacles using slaves and criminals. The latter did not always involve deaths, but reenactments were a popular way of dispatching hundreds of condemned men at a time. For executions, it is impossible to give a definite number but sometimes several thousand captives, runaway slaves and criminals were executed during an event. Thus, the estimates given by various sites of a total (gladiators + executions/spectacles) of 400,000 to 700,000 people killed in the Colosseum do not seem unreasonable. The large majority of these would have been criminals, captives and condemned slaves rather than trained gladiators.
Source:The Colosseum in Rome
To the above must be added the numerous deaths in ancient Rome from the city being sacked several times, and that political upheavals/purges (e.g. Sulla's proscriptions) and riots also led to large numbers of deaths. Unfortunately, establishing how many of these were in a specific area is, of course, impossible. Finally, we need to consider the number of homicides in an area (within one square mile of the Colosseum) that has been populated for at least 2,200 years but, again, it is impossible to give a number (though the homicide rate in ancient times in nearby, densely populated Suburra was reckoned to be very high).
The Colosseum itself occupies just 6 acres out of the 640 acres in a square mile. Even if there have been square miles with more deaths by conventional weapons, it is unlikely that these 6 acres can be surpassed anywhere in the world. The scale of the carnage was such that disposing of the bodies (human and animal) was a major logistical problem.
If the scale of these deaths is hard to believe, never mind comprehend, consider the following from Keith Hopkins, formerly Professor of Ancient History at the University of Cambridge:
Rome was a cruel society. Brutality was built into its culture in
private life, as well as in public shows. The tone was set by military
discipline and by slavery. The state had no legal monopoly of capital
punishment until the second century AD. Before then, a master could
crucify his slaves publicly if he wished. Seneca recorded from his own
observations the various ways in which crucifixions were carried out,
in order to increase pain. At private dinner-parties, rich Romans
regularly presented two or three pairs of gladiators: 'when they have
finished dining and are filled with drink', wrote a critic in the time
of Augustus, 'they call in the gladiators. As soon as one has his
throat cut, the diners applaud with delight'. It is worth stressing
that we are dealing here not with individual sadistic
psycho-pathology, but with a deep cultural difference. Roman
commitment to cruelty presents us with a cultural gap which it is
difficult to cross.
Although animals are definitely not within the scope of the question, it seems appropriate to at least mention that the killing of beasts was on a scale most likely unprecedented in history. Such was the Roman thirst for animals to kill in the arena that numerous species disappeared from North Africa.
Note: one answer on Quora gives 3,120,000 people killed in the Colosseum, but this is based on a misinterpretation of Hopkins & Beard's calculations.
Gregory S. Aldrete, Daily life in the Roman city : Rome, Pompeii, and Ostia
6 Infamous Sacks of Rome
The dangerous streets of ancient Rome
10 Fascinating Facts About the Roman Colosseum
Facts about the Colosseum
R. Laurence, Roman Passions: A History of Pleasure in Imperial Rome (2009)
The Roman Theater of Cruelty
Body Count of the Roman Empire