The only time a private individual may take the life of another is in order to defend the lives for which he is responsible. And, as the subsequent verses in Exodus 21 go on to clarify, this was not to be a literal knocking out of teeth or gouging of eyes, but rather a reminder to judges to punish fairly and proportionately. This law of proportional punishment was actually meant to prevent personal vengeance, by specifying that the penalty for crimes must be exacted only by the civil authority, and be no more severe than warranted by the crime. Christ was not renouncing the Old Testament standard of judgement in His statement, but issuing a stiff rebuke to the religious teachers of His day who had twisted this law and applied it to personal slights in order to justify their own vindictive desires. And if he is thirsty, give him water to drink.
Personal vengeance is forbidden - in both Testaments.
In the context of the Sermon on the Mount it seems clear that Christ is addressing the heart attitude of a person towards those who mistreat him, rather than making a categorical statement about all use of physical force. This means we should have regard for the true good of people rather than simply to give them what they say they need or to yield to their every demand. Both are tools in human hands, and each is capable of destroying, or furthering, life. Both oxen and firearms are tools to facilitate life - albeit tools with very different functions. To our modern way of thinking, the best way to deal with the potentially dangerous ox would be to ban oxen altogether.