Some people did, but many others did not. The fourth and fifth centuries were not fun times to live in, but the Roman Empire at its height wasn't a great place either. A good example is the Sack of Rome in 410, an event often seen as leading up to the eventual end of the Western Roman Empire, even though it was 67 years before the deposition of Romulus Augustulus. Contemporaries were surprised and shocked at the city's sack, with figures such as St Jerome exclaiming that:

Who would believe that Rome, built up by the conquest of the whole world, had collapsed, that the mother of nations had become also their tomb.

However, we have to recognise that rhetoric does not represent reality. As far as sacks go, the Sack of Rome was a relatively mild one, and although we have some evidence of public unrest in Carthage in its aftermath, many people were level-headed enough to address the public's fears. In this case, St Augustine rose to the occasion and delivered several sermons in Carthage, arguing that the Sack did not herald the apocalypse - the empire was man-made and entirely fallible, so even if it was destroyed, it did not mean that God was giving judgement on mankind already. Augustine continued to expound on this theme in his City of God and also commissioned his friend Orosius to write a History Against the Pagans to look for historical examples of calamities affecting the empire; if disasters had occurred in the past and the world did not end, why would the fifth century be any different?

Moreover, Rome was no longer the political centre of the empire - the senators who lost their wealth in the Sack were generally not those who were involved with the imperial administration, as these powerful men were serving in the imperial court, where real power was, or in the provinces. The Eastern Roman Empire and Constantinople for example were doing very well even as their western counterparts struggled to survive. Although it was embroiled in various internal political struggles and had to pay off the Huns repeatedly, economically it was booming and we find no evidence of clergymen engaging in debates about an impending apocalypse in fifth-century ecumenical councils. From our perspective, the councils' focus on Christology and personal conflicts seems trivial, but at the time monks and bishops were literally willing to fight over these issues, often involving the emperor in their disputes. These are the not the actions of people dreading the end of the world, but the actions of those trying to preserve doctrinal purity in a bleaker but still active world.

In any case, many of the Roman super-rich living in Rome in 410 had survived and taken refuge elsewhere in the aftermath of the Sack. In fact, within a few months Jerome would be writing to the prominent Anician family in North Africa, allegedly to instruct one of their daughters in asceticism. At the same time however Jerome was clearly angling for some form of patronage from the Anician women, for he was experiencing financial trouble at the same time. Even for Jerome, there was no real sense of impending doom that actually affected the way he lived his life.

As I said, the Western Roman Empire wasn't in the best shape and we have a lot of evidence of economic decline in this period, but to what extent could this be seen as evidence of world-weariness on behalf of the population? Not very much I'd say. Even in a time of political crisis, for many people life went on as always. Sometimes communities were ravaged by war and plague, others were left alone or had to pay their taxes to another master, but these were problems that humanity had dealt with throughout history. Some people did indeed treat this as signs of the apocalypse, but there was also a varied and exciting dialogue between articulated individuals throughout this period, so to describe the fifth century as a century of decline would be ignoring both the vitality of Roman culture and be imposing a narrative on a non-teleological process. The Western Roman Empire repeatedly looked like it was going to survive the various crises it faced, for otherwise many people would not fight so hard to defend it. When the Western Roman state finally collapsed in 476, many people in the western provinces had already been living in a post-Roman world; the empire ended not with a bang, but with a whimper, the furtherest thing possible from a world-ending crisis.