It is difficult to say how they would have interpreted their finds, but there were definitely some ancient "antiquarians":
According to various sources, the Neo-Babylonian king Nabonidus (ca. 6th c. BCE) was an avid collector of antiquities. His daughter, Ennigaldi-Nanna, curated the collection of historical artefacts-- excavations have yielded some of the earliest known artefact tags.
Although they were common some two and a half thousand years beforehand, libraries arguably became "fashionable" with the rise of the cosmopolitan Hellenistic age (after the death of Alexander the Great in 323 BCE). It was a great accomplishment for cities throughout his fractured empire to have vast collections of historical and theological texts, among others. The most famous of these libraries may be quite familiar to you. Some of the texts that may have been stored at these libraries would be familiar to historians today, such as the works of Herodotus.
This tradition carried into the Roman period. Although the Romans are often labelled as ruthless conquerors, they also held enormous reverence for cultures older than their own; it would seem that they felt there was a lot to learn about successfully running an empire. This is in part why they were often keen on "escorting" certain deities from a conquered area into their own pantheon: Juno Regina, for example, was brought to Rome from Veii after she "allowed" for her city to be taken by the Romans in the 4th c. BCE. They repaid the Etruscan goddess by constructing temples on the Field of Mars and the Aventine Hill.
The interest in the past continued into the Christian period, too. When Constantine came to power, he provided his mother, Helena, with funds to travel to Jerusalem and locate the cross on which Christ was crucified. This is why Helena is now considered to be the patron saint of archaeologists.
Finally, please note that this is only an analysis of a tiny slice of the world. I'm sure there are some folks here who specialize in the histories of East Asia, India, South and North America who could provide examples from their areas of specialization as well :)
As a pernickety side point, "archaeology" is often reserved for the systemised academic discipline that emerged in the late 19th century. People with an interest in the material remains of the past before that point are usually called "antiquarians".
Thutmose IV (eighteenth dynasty) claims that while he was out hunting, he found the Sphinx, buried up to its neck in the sand. This is both a little surprising, in that apparently the Egyptians didn't care enough about their own sphinx to dust it off, while not being very surprising at all, considering the statue was about a thousand years old when Thutmose came across it.
Anyway, he took a nap in the shade cast by the statue's head and, while he slept, the sphinx came to him in a dream, saying that if he unearthed its body, it would make him king. He did and it did, according to a stele (carved stone tablet) left behind commemorating the event.
Khaemwaset, a son of Ramses II was responsible for restoring numerous ancient (even to him) sites during his life, many of them from the fourth dynasty, making them, again, about a thousand years old at the time. Why he was so interested in these things isn't clear, but "isn't clear" can pretty much describe anything having to do with Ancient Egypt.
That these men did obviously isn't archaeology, but it does show that a) the 'recent' ancients (circa 1400BC) were very aware of the relics of the past, b) those relics held power over their thinking and c) they were willing to put considerable effort into maintaining the past, at least sometimes.
As an add-on question, what kinds of museums existed in this era?
Self follow up question: what about China and India? Would, say, a Chinese archaeologist in the year 0 have been digging up 2,000 year old relics of his forebears?
People in the Early Anglo-Saxon period (5-7th centuries in England, right after the Western Roman Empire collapsed) were very interested in the history that surrounded them.
They collected old Roman objects, probably as novelties or trinkets (though sometimes you find things like Roman spoons and old Roman pots in settlements which were being used - grandma's silver and china, perhaps?). Sometimes these antiques were re-worked into jewelry: coins with holes punched in them for necklaces, or big bracelets turned into finger rings. Often, these objects are very worn - the coins have frequently had their faces rubbed off, which means people touched them a lot. Perhaps they rubbed them for luck, or played with them when they were feeling fidgety.
You often find these curated antiques in people's graves, buried in pouches on the belt or else tossed into the grave (rather like we toss lucky pennies into fountains, perhaps). Some scholars think there was a magical element to this: were old Roman objects used as amulets for protection? A recent article (published last week!) has noted that these objects were more often buried with children, and has suggested that the loss of a child reminded people of the other things they had lost, like the prosperity of the previous age, and they buried reminders of these losses in these children's graves as they grieved.
Where did they get these antiques? They must have done some scavenging. There's good evidence that people raided Roman ruins for raw materials, and while they were there they probably kept their eyes open for other buried treasures. The way in which they curated these finds - keeping them, collecting them, and burying them with the dead - suggests that they probably recognized these as historical objects the same way a modern collector would (though one article suggests they didn't recognize them as Roman - unlikely, in my view).
At the same time as they were collecting antiques from old Roman ruins, people were also investigating prehistoric sites. Cemeteries were often built next to, or on top of, 2000 year old Bronze Age barrows. At first, these graves were dug into ditches around the old mound - getting as close as possible. By the seventh century, they were putting important people right in the middle of the old mounds. Howard Williams (1) argued (2) this was because these people wanted to connect themselves to the ancient inhabitants of the land: they sought to overwrite the ancient past with their more recent ancestors, to stake a claim to ownership that no on could argue with (see also). This is almost archaeology in reverse: digging up the past to erase it, instead of to study and protect it.
The ?8th century poem Beowulf tells us a little bit about who they imagined these ancient barrow builders to be:
long ago, with deliberate care, somebody now forgotten had buried the riches of a high-born race in this ancient cache. Death had come and taken them all in times gone by and the only one left to tell their tale, the last of their line, could look forward to nothing but the same fate for himself: he foresaw that his joy in the treasure would be brief.
A newly constructed barrow stood waiting, on a wide headland close to the waves, its entryway secured. Into it the keeper of the hoard had carried all the goods and golden ware worth preserving. (from Heaney's translation, lines 2233-2246)
In some ways, you can already see the original identity of the barrow builder starting to be erased: as a lone hero, last of his kind, carrying his treasure into the ground, he foreshadows Beowulf's impending death. But it's a great passage, because it tells us that people in early medieval England were actively thinking about the archaeology that surrounded them, and imagining stories to go with the prehistoric ruins they saw. It's no scientific archaeology, but it's a real interest in the past - and if they knew there was treasure inside some mounds, that means they were definitely doing a fair bit of digging!
As /u/brigantus mentioned in this thread, Archaeology as a way of learning about the past is a relatively recent discipline.
This is not to say that ancient peoples were not interested in the past or its material remains. Some of it was pretty hard to ignore, like the pyramids /u/crow_hill mentions. Without the systematic study of material remains (ie. Modern Archaeology), though, they had to be a bit more "creative" to fill in the gaps in their understand of the distant past, relying heavily on oral tradition, religion, folklore, and previous historians' accounts.
In this thread, I talked a bit about how a particular Roman historian interpreted prehistoric remains from the Bronze Age Greece through myth, rather than systematic study like contemporary archaeologists.
In the book "Politics" by Aristotle he mentions the "Earth People" people who lived on the outskirts of society, dim witted, and smaller than the standard Greeks of this time. He also references the Gauls and other tribes from Europe that predate his version of modern Greece. But like most progressive civilizations, its not about who you conquered and erased, its about who you are GOING to conquer and erase.