We don't consult the manuscripts themselves, and, in fact, most modern historians would not be able to read the manuscripts even if they had them in front of them. They are pretty hard to read. edit: here is a fun page about deciphering medieval/Ren. manuscripts. Fortunately for us, the texts have (mostly) been painstakingly studied and transcribed for us into legible, convenient texts we call "editions." A very common go-to edition for most Greek and Latin texts is the Oxford Classical Text (or "OCT"), which contains the text itself in a neat, modern, readable font as well as an apparatus criticus at the bottom of each page which tracks discrepancies in the various manuscripts for that section.

If for some reason we needed or wanted to examine a manuscript directly, then yes, we would go to the place where that manuscript is held (after asking for permission and arranging it all, of course—one does not just drop by to access a priceless manuscript). This is very rare, unless the person specializes in paleography or preservation or is working with something exceedingly obscure.

Many, but not all, manuscripts are already digitized or are in the process of being digitized. There are various search databases, none comprehensive as far as I know. manuscriptsonline.org, for instance.

For other types of ancient documentary evidence, there are generally databases. For papyri, there is the wonderful papyri.info, which draws on several papyrus databases like APIS. For Epigraphy, there is (for instance) the fantastic PHI search (epigraphy.packhum.org).

I feel the other answer is incomplete, because it doesn't cover all types of "ancient" manuscripts.

As stated, most ancient Western manuscripts were long ago published as books on printing presses, generally between 1500 and 1900. The various publications and other manuscripts were then compiled into critical editions so that the differences between manuscripts could be compared; this mostly happened between 1800 and 2000. After 2000, these have increasingly been digitized, as the other answer stated. A lot of work can be done now via Internet databases. Some lesser-known medieval manuscripts have not received editions yet, but these are generally the ones that are not used so often in research and historical writing.

East Asian medieval manuscripts of ancient or classical texts were also published as books over the same time frame, but not all of them became critical editions or were digitized. It is often still necessary to go get the physical books. (Japan and South Korea have much better digitization programs than China, but even in that case, physical access is often helpful.) If you are writing a dissertation on premodern East Asia, you will likely accumulate a big library of flaky old books. I do not think it is a bad situation for historians, just one that will require a bit more space.

Indian manuscripts have often never been published! Many pre-colonial and early colonial manuscripts are still preserved only as physical one-of-a-kind documents in poorly funded libraries. Because of the climate of India and the lack of interest in building and maintaining climate-controlled archives, this can be a quite perilous situation and as of 2022 we are probably losing a few manuscripts forever every year or so. I don't know the situation for archival work in Pakistan, but I know there are pre-colonial archives there and I can only imagine a similar situation.

In Southeast Asia, the climate is even worse for writing materials, and most "ancient" manuscripts are long gone. There is not much to consult from pre-1600 ancient times, even if you want to. However, precisely because there is not much to collect, the collections that exist seem to do a better job. For example, rare Indonesian manuscripts are generally preserved either at the University of Leiden or at the National Library of Indonesia, so we are not as likely to lose them as we are rare works from India and Pakistan. I think basically all of the medieval ones have been published now, too, although not digitized. (Rather obnoxiously, a lot of ancient Burmese manuscripts were sent to Indian libraries in colonial times, and are now very hard to access and in unclear physical condition; Michael Aung-Thwin has commented on this.)

edit: Aw, I forgot to mention Arabic manuscripts! I don't know the general situation with these. They are generally medieval or early modern texts, not ancient in origin, and I think the climate is often better but they have been threatened by war. For example check out the Timbuktu manuscripts, a great treasure of medieval sub-Saharan Africa, which were saved from an Islamist uprising, only to suffer from the humid environment in southern Mali, so that they had to be sent back to the inaccessible desert town of Timbuktu after the uprising ended. Also look at the manuscripts in Yemen, which have been secretly digitized under cover of darkness during the civil war after 2014. Yemen is basically a natural fortress impenetrable by imperial armies, so some of its medieval manuscripts preserve parts of Islamic history and philosophy otherwise lost to us.

I disagree with some aspects of u/Alkibiades415 response - in my experience, graduate training in the US focuses heavily on access to new primary sources that often require trips to archives where (in certain fields) the historian is one of the handful of people to have ever read a source. For example, a professor of mine described reading 17th century texts where the clerk had sifted sand between each page of a ledger to keep them from sticking, and every time he turned a page the undisturbed sand would trickle off onto his desk.

The reason for this is that there are enormous quantities of historical writing, and except for the most obviously important sources, there is not the time, talent, or money available to digitize them, let alone translate them. Consider for example the archives of the Dutch East India company, whose 25 million pages take up 1.2km of shelf space (https://www.nationaalarchief.nl/en/explore/the-world-of-the-dutch-east-india-company). Some of it is available online, but most has never been read by a modern reader. Picking examples more or less at random, other important sources like the 150,000 letters and correspondence of the medieval Italian merchant Francesco Datini have been digitized but not translated (http://datini.archiviodistato.prato.it/en/il-progetto/). To read them, scholars learn pre modern Italian and rely on handbooks that serve as guides to early writing styles (see https://archive.org/details/CappelliDizionarioDiAbbreviature). Similarly, scholars of 20th century topics often use collections of letters and papers that are not always scanned. In these examples, a scholar would have to go to the specific archive (or the website of they are available online) and go through untranslated material that is often not indexed. In essence, you just start requesting volumes of bound papers or boxes of material that are from the period that you are interested in and start looking for interesting material.

While, as u/Alkibiades415 decribes, there are many standard published editions that provide excellent translated editions of important manuscripts, as well as other websites and digital repositories that provide portals for certain classes of sources many historians seek out new materials that require visits (whether virtual or in person) to sift through a collection at a particular archive to look for useful information and insights.