This is really more of an archaeological question than a purely historical one. Reading the poem The Ruin got me wondering what the ruins would actually have looked like shortly after they were abandoned.
Today many Roman ruins have had to be dug out of the landscape to make them visible (I'm thinking of somewhere like Vindolanda). But what kind of condition would forts, aqueducts etc. have looked like in, say, the year 800? Would they still be largely functional? Or did the buildings degrade quickly?
Some may have deteriorated depending on weather conditions, etc. but a lot of former Roman buildings would have been reappropriated and turned into something new. For instance in Brescia there is evidence that people took up residence in one part of the old Roman buildings in the 6th-8th centuries.^1 Trier would be another good example of this, mainly the Constantinian Basilica (which is still standing to this today and is relatively untouched). A lot of Roman buildings would still be in use from the looks of it, although certainly some would have been abandoned or modified. Further, some may have been made worse by medieval people taking building materials from them (the amphitheatre in Trier is another example of this), there is an entry in the Liber Pontificalis which reports how the roofs of some of the buildings in Rome had been taken by the Byzantine emperor. Or in Ravenna Charlemagne took some of the marble columns and placed them in his palace at Aachen!
Additionally, you might want to check out Michael Greenhalgh, The Survival of Roman Antiquities in the Middle Ages which discusses a lot of different things you would probably be curious about.
A follow-up question, if I may. In the show Vikings, the anglo king shows Ragnar Roman artefacts. IIRC he tells him that people believed those ruins to be work of a mythical race, because they couldn't fathom the idea that Pagans could be more advanced than Christians.
Is there any truth to that? Did the common people gradually forgot who the Romans were, what they did, and why? In the show they make Romans look like some kind of Precursor race as in many scifi works.
It really depends on use. Most sites contracted as populations turned to dispersed settlement and agriculture. For example, Wroxeter in Shropshire, in the British Midlands, was one of the biggest cities in Roman Britain but by the 9th Century was largely abandoned and in areas ploughed over apart from one small area.
9th Century Chester was, according to the Anglo-Saxon Chronicle, entirely desolate and in ruins, but Æthelflæd 'restored' the walls in 907 and the amount of jewelry, coinage and decorative metalwear found nearby shows that the city was flourishing and grew incredibly quickly. Many of the Roman towers and buildings were restored, and their restoration is celebrated in a series of pictoral coins minted between 907-920.
Some Roman signal stations are repurposed as fortresses and as part of a Vigilis Marinis from the mid-9th Century onwards, and the roads and bridges tend to be maintained as part of the trinoda necessitas which become standard form on Anglo-Saxon charters from the 8th Century onwards. As a rule, sites and buildings which are militarily useful tend to be maintained, or abandoned and later restored, but more civic structures tend to be crumbling into disrepair by the 8th and 9th centuries unless they're in resettled areas.
Are you thinking of any particular region?