It depends wildly on the archive holding the material. The easiest to use is digitized material which requires only a computer, internet access, and knowledge of where to look.

National archives, like those of France in Paris or the US in Washington DC are often some of the easiest to access- they are public institutions and people use them all the time to research genealogies or family histories. You might have to pay for an ID card and take a tour but there those barriers aren't usually too hard to overcome. The difficulty might be figuring out exactly what you want to see and what might be useful for you-- there is a lot of archive material out there in the world, the tax receipts for that French town might be several cartons at least of material.

So general access can be easier- but the older the document the more likely you will only be able access microfilmed copies of it, the less likely you would be to get to see the original without connections or reputation as a historian. And this becomes more true in general for smaller archives, public access might be limited, they might require reservations or letters of recommendation. Smaller regional or provincial archives or museum archives can rather jealously hold their material understanding full well that the more it's accessed the more it can deteriorate. Researching a ships logs at a museum archive once, I have to wear gloves and use to tweezers to handle a copy (not even the original) of the logs.

All archives are different though, so it really depends on what you want to see and if you can figure out what archive might have it. Sometimes archives have finding aids-- sometimes they don't really and you find yourself rooting around in a carton of misc. government records looking for a file or letter that may or may not be buried in there. When things are catalogued or organized and finding aids or descriptions made, it's easy, when those don't exist you're sometimes just shooting in the dark.

If you are enrolled in a good university, then they are usually subscribed to plenty of journals, databases and archives that you can access. That is how I access my material. Really, being a part of an institute that subscribes to hundreds or thousands of sources is the only practical way to gain access to some of these materials as the subscriptions can be very costly.

If you can't access a journal or archive because you are not at a university, you might want to try a library (particularly State or National Libraries). Depending on what they are subscribed to, you might find Journal articles analyzing historical murals, frescoes or graffiti.

Many national libraries and archives have a variety of sources you can access. For example, I have been reading various French Medieval manuscripts that the British Library has digitalized for free. There are also reprints of some medieval manuscripts which are really useful if you want to actually see the illuminations.

It also depends what you are interested in, I know that certain libraries have genealogical records and that you may be able to access the records of a European town, particularly say if you are after a copy of an ancestors birth record (although be prepared to pay a fee for that).

It can depend entirely on the source. Some libraries have open stacks, some archives just require you to speak to the curator or archivist. It helps to have academic credentials to open doors, in large part because they assume you've had training in handling the more fragile sources.

Honestly, just asking is your friend. You can find out what the options are for what you want to see. Even if you can't see something directly, they may have a print or online edition available for your use.