I want to differentiate between manuscripts and papyri. There are no ancient manuscripts—the oldest come from about the 9th century CE and are themselves copies of copies of copies of copies of copies. But yes: from about the 9th century to the early modern period, hundreds of manuscripts survive. They are housed in rare document libraries all over the world. Many are located in the Louvre, in the British Museum, and in the Vatican. A person can access them, but as you can imagine, they are very rare, fragile, and precious, and not just any random person off the street can stroll in and handle them. You need special permission and you need to justify why you are there, what you are looking for, and without academic credentials or celebrity status, you will probably be denied. At the Vatican, for instance, for a person with proper credentials and permissions, the process of accessing rare manuscripts can either be fairly simple or incredibly convoluted, depending on who you deal with. To my knowledge, no one except Vatican officials are allowed down in the vaults: they bring your requested manuscript to you.

Now, there are ancient documents: papyri. There are hundreds of thousands of fragments in the university libraries of the world. Oxford alone has hundreds of thousands of fragments, most tiny little shreds but many substantial pages as well. Berkeley, Yale, Michigan, the British Museum, Chicago, Columbia, Duke, Penn, Princeton, NYU, Stanford, Wisconsin, etc all have collections (apologies for those I forgot to list). These also are very precious and cannot be accessed by strangers off the street. Some of the larger and more famous fragments get conserved and encased in glass for viewing, and these can be accessed without too much hassle, if you really want. Most of them are in ancillary buildings and collections, not the main library. Of course, none of these places are open right now, due to the pandemic.

A great example of a repository of rare documents is Yale's Beinecke.

I have experience working with manuscripts in the Bodleian and the Cambridge University Library, as well as at several of the college libraries at both Oxford and Cambridge. I was looking for very specific manuscripts for my doctoral work (different copies of the same text), and I was looking at them from the standpoint of interest in the text itself rather than the illumination. All of the visits were arranged well in advance, with specific requests made according to the regulations of each library. (Cambridge and the Bodleian have all of their requirements online). I had to make sure that I could get in to see the manuscripts on specific days and that the staff would be able to have things ready for me. (That was actually a challenge at Cambridge that had me rearranging my schedule because I had planned to be there right at the beginning of the fall term, when the staff would be busy with all manner of new student issues.) I was in no library just allowed to "browse the stacks"--the manuscripts were fetched for me.
Standards for handling the manuscripts varied considerably--about half of them required gloves; the rest did not. There were also weighted cords used to help hold down pages without touch. Since I was a doctoral student, it was generally assumed that I knew what I was doing around manuscripts and would treat them properly. None of these were particularly famous manuscripts, so the mere fact that I could contact these libraries with interest in a particular text in a particular manuscript was a good sign of my bona fides, but I also always included a description of the project I was working on. (For those manuscripts I ended up choosing to focus on for my project.

Over on the other side of my interests (as a calligrapher and iluminator): Many years ago I also was part of a group that was able to get a private viewing of some of the Pierpont Morgan's most well-known works of illumination (including the Maciejowski Bible). For these, we didn't actually touch anything--the person giving us the showing did all of the holding and page turning (and absolutely wore gloves.) I suspect this would be the case for any of the more celebrated illuminated manuscripts, unless you were specifically a manuscript studies specialist.

Personally I've only worked in the Bibliothèque nationale de France, but the process is generally the same as in the other answers here - I needed to prove my academic credentials, and I needed a reference letter. I brought a letter from my boss in my post-doc program. After verifying everything, they gave me a library card (valid for 6 months I think).

Then, the BNF (like other libraries, as far as I'm told) prefers that you look at the manuscripts on microfiche, if they've already been scanned. I had to go to the microfiche room first, but it's like a game really: once you're there, you need to tell them that the microfiche is unclear, or you need to examine the physical manuscript for codicological reasons or whatever (like, you want to measure the size of the book, inspect the parchment/ink, etc). No one wants to look at microfiche, it's all very silly.

So then you can go to the manuscript reading room and you fill out a card with all the details of the manuscript - you really need to look this up beforehand, because the info they need to get the manuscript for you is not necessarily the common name of the manuscript, or even the common shelf number. For example the one I wanted to see is generally known (to me anyway) as "Fr. 19026", but it's actually called something else if you want them to bring it to you. All part of the game I suppose. At last, they'll bring you your manuscript and you can read it in the reading room. You never get to see where they actually store it though!

For medieval manuscripts you don't actually need to wear gloves. Medieval historians who work in archives can get very, er, passionate about this. For me, seeing a historically inaccurate movie is annoying, but for people who do lots of archive work, they get really mad about people on TV wearing gloves. Here's some info about gloves from the British Library:

https://blogs.bl.uk/digitisedmanuscripts/2011/08/white-gloves-or-not-white-gloves.html

Lastly, the most important rule when visiting an archive: no pens! Don't even think about bringing a pen in there!

Yes! the United Kingdom National Archives at Kew Gardens is completely free and accessible to the public- you should be able to arrive there, get a reader card (for free) and then request the documents you would want to read. I don't know about 'ancient texts' or white gloves (though they have very strict rules on food and drink), but when I was studying my MA I was able to read copies of original war diaries of the Irish Guards there from Operation MARKET GARDEN, as well as the originals of several other documents, such as briefing papers from the cabinet office, operational planning resources and other bits and pieces.

You can learn more about them via their website and search their catalogue of pretty much anything that might be of interest to you.