It varies a lot. Later European archaeology is generally quite well-founded as it ties in to a historical record. While I don't have any familiarity with the Colosseum, I have heard that there are records of payments to sailors for manipulating the shades.

If you get further away from historical records, it does get more tenuous, but there is often a solid basis of scientific or technical evidence. My own doctorate is in use of physics for dating by thermoluminescence and optically stimulated luminescence, which together cover a lot of ground for stuff that cannot be dated by C14. There are several quantitative techniques like that, for instance for isotopic analysis which can be used for deducing trade routes. There are also many qualitative and semi-quantitative techniques, varying from traditional trenches and ground penetrating radar to pollen analysis (which gives you information on climate and ground cover).

However all of this has to be tied together in to a picture of human culture. That necessarily has a large element of interpretation. This is not unique - you can see the same pattern in fields like astronomy. Generally I think archaeologists are reasonably good at knowing when they are on uncertain ground. If I may give an example: in Scotland, mainly on the north west coasts, there are the remains of pre-historic buildings called brochs. The biggest stands 40' high, and it is possible to walk up through the stairs between the double walls to the top. We know roughly when they were built, but not why. There are several theories, but I find it interesting that there is an absence of dogmatic assertion that one particular interpretation is true, and it is still possible to have a sensible discussion about a "fringe" theory (my own idea is that some of them may have been for excarnation) which will centre on what evidence you might need to search for (bone fragments in this case).