Contrary to popular misconception, Medieval people definitely did have a sense of the past, though one somewhat different to our own. They certainly didn't hold to a progressive view of history, in which societies get better over time - indeed, it was much more common to see society as having declined from a past golden age though, that being said, most people who held to such views had some kind of political or moral axe to grind i.e. Dante Alighieri's disaffection with the factional politics of the Italian city-states resulted in him longing for the golden age of the Roman Empire under Augustus in which the (Mediterranean) World) had enjoyed lasting peace, as expressed in both his Divine Comedy and his political treatise De Monarchia. To give another example, John of Salisbury, in the 1150s, argued that knights should try to emulate the discipline and loyal service to the state of Roman legionaries.

They certainly did have a sense that the past would have looked and felt different. Obviously, archaeology hadn't gotten very far (i.e. Pompeii would not be excavated until the 18th century) though we do see some proto-archaeological work going on. For example Guibert de Nogent (1055 - 1124), in his history of the Abbey of Nogent, is confident that his abbey was built on a pagan burial ground because "vases have been discovered in these tombs the likes of which are unknown in Christian times. I can therefore find no explanation except to think either that these were pagan tombs or very ancient Christian tombs arranged according to pagan customs." Or, to take a different example, the great sculptor, Nicola Pisano, when he was making the reliefs for the pulpit of the baptistery at Pisa Cathedral in the 1250s, looked at Roman funerary monuments in the ancient cemetery near Pisa, and so depicts the Virgin Mary like an Augustan era Roman woman.

The reason why so many works of medieval art portray people from Classical/ Biblical times like contemporary figures is in part because, as I've said, archaeology hadn't gotten very far and in part because they wanted to create figures that they felt could be relatable to people in their own day. For example, 15th century tapestries portray David, Hector and Julius Caesar as contemporary figures because they were meant to serve as role models to knights and kings.

Edit: I just re-read the question and you state pretty clearly how did people understand long periods of time is your question. I'm gonna leave this up, but mods, please remove if it breaks the rules or anything.

I'm not sure if you mean "time" to be "when is it 4 o'clock?" or "the ruins of the Colosseum have shown that time has passed", but I can do my best to answer the first.

E.P Thompson has an essay called "Time, Work-Discipline, and Industrial Capitalism" from 1967 that explores this. In this essay he posits that most people started to use "clock-time" when wage labor became popular. You need to know exactly how much time has passed when you work for so many dollars per hour, so exact timekeeping becomes important. Of course, Thompson has to explain how time was kept prior to this. He refers to this as "task-oriented time". Simply put, you measure time in how long it takes to complete a task. A modern analog might find a dialog like this:

"How long until dinner?"

"The length of a Game of Thrones episode"

You don't know the exact length of an episode of prestige TV (in minutes), but you have a pretty good idea of how long it will be until dinner is ready.

Thompson looks at contemporary anthropologists and their findings on time to give examples of how time is kept without clocks. Cooking time is generally constant and is therefore a reliable measure.

In a similar way terms evolve for the measurement of time intervals. In Madagascar time might be measured by "a rice-cooking" (about half an hour) or "the frying of a locust" (a moment). The Cross River natives were reported as saying "the man died in less than the time in which maize is not yet completely roasted" (less than fifteen minutes). (58)

This task-based time can be effective in that you have a general idea of how long things take. An alternative method of time keeping may also be used, such as the "cattle clock" discussed by E.E. Evans-Pritchard:

Among the Nandi an occupational definition of time evolved covering not only each hour, but half hours of the day - at 5-30 in the morning the oxen have gone to the grazing-ground, at 6 the sheep have been unfastened, at 6-30 the sun has grown, at 7 it has become warm, at 7-30 the goats have gone to the grazing-ground, etc. (58)

Even though there aren't exact time values associated with this, you could arrange a meeting for "when the goats have gone to the grazing ground" and someone would understand what time you want to meet.

In terms of "real" clocks, Thompson looks at church bells that are supposed to be rung at specific times, but he doubts the accuracy of these:

From the fourteenth century onwards church clocks and public clocks were erected in the cities and large market towns. The majority of English parishes must have possessed church clocks by the end of the sixteenth century. But the accuracy of these clocks is a matter of dispute; and the sundial remained in use (partly to set the clock) in the seventeenth, eighteenth and nineteenth centuries (63)

Eventually clocks as we know them begin to come about:

A great advance in the accuracy of household clocks came with the application of the pendulum after 1658. Grandfather clocks begin to spread more widely from the 1660s, but clocks with minute hands (as well as hour hands) only became common well after this time." As regards more portable time, the pocket watch was of dubious accuracy until improvements were made in the escapement and the spiral balance-spring was applied after 1674. (63)

If you want to know more about clock engineering and its advances I'll let you read the rest. There's also a very informative section on when watches began to be used commonly and how it coincides with watch production more generally.

Hope this helps!

EP Thompson - Time, Work-Discipline, and Industrial Capitalism