It'll depend on which society you're performing your internet archeology from, and what information that society intends to preserve, especially if data types have to be converted to be forward compatible, or have tools that are backwards compatible as necessary.
A lot of the other answers have discussed the hosting aspect of a site, but I'd like to approach this from the worldbuilding consideration of which society is the one you're viewing the internet from.
Some sites might still be up, technically, but if your internet archaeology is from other countries than the one where the content is hosted, you may run into outright blocks on the content you're attempting to gather archaeology of. This video from the Patriot Act with Hasan Minhaj about Saudia Arabia + Censorship in China is an easy example of some of the issues with regards to censorship of internet properties or content making it hard to view them from an archaeologist approach, but also skims a little bit over the case of a workaround, known as a VPN (Or Virtual Private Network).
VPNs here would allow a user to view a website as if they're in another region (i.e. I personally can't watch Last Week Tonight videos uploaded to YouTube on their channel immediately, as being in Canada, they're blocked for ~2 weeks, but if I VPN'd into the U.S., I would then be able to view them when they were initially uploaded), provided that region still exists, and has a VPN that you can connect to in their region.
In a society post our current one, if, say, a localized Mad Max situation localized to the U.K., no longer has computers on its land due to an extinction event (That probably removed people too, or people tied to the culture of the U.K. in this example.), if Netflix still has the licensing setup for the U.K. to have U.K. specific content, someone in the U.S. would need to travel to the U.K., set up the VPN connection and subsequent internet cabling to allow for a remote connection to Netflix, and then attempt to view it that way, if you wanted to see how the U.K. had been able to view Netflix, and perhaps see what was Trending before the Mad Max situation was involved.
But, okay, those are extreme situations - that's a localized societal collapse view of archeology of the internet, and ideally, we're thinking these computers are still up and running, and their information can be accessed, perhaps miracle-hosted by a few Internet Archive style groups. Surely the technology will be kept up to date and accessible?
That requires Unicode to continue being the standard for transcribing texts online (Specifically, that it'll work in UTF-8, or at least that UTF-8 will continue to be supported in the future.), and that old documentation won't be replaced as they continue to add new codepoints and specific characters, if by chance, they run out filling spaces for Wingdings to be supported by the Unicode Consortium. That's unlikely to change anytime too soon, but if we're looking at attempting to preserve internet content across every website regardless of if someone is willing to update their libraries, then the concern of forward compatibility in those documents could be of concern. Unicode itself is relatively new, and there were quite a lot of other codepoints that needed to be put together at that point that were subsets of other character sets and their ordering of characters, that to expect it to stay absolutely locked in place is unlikely. At which point, you run into issues like the one described by the Wingdings addition (Specifically, if some characters did not render correctly, they would instead jump to an unrelated character - like a "smiley face" Wingdings character into a "J" ASCII character.) In a Mad Max world with technology still around but without international internet cables, there may be a drive to develop devices that specifically jump back to older, more limited character sets, thus ruining the ability to really be able to read the documents.
Okay, that's not great, but at least the presentation will stand up for the document, right?
This is more of a minor detail than the others, but if you're looking for them to stay the same look and feel, on occasion, HTML document elements have been known to be deprecated.
Depending on how a specific site is developed with CSS components expected in question, the deprecation of a feature there could cause documents to fall out of the regular presentation as well. Admittedly, this is cherry picking at the moment, but could be more widespread after a few decades, or centuries, of internet developments being made.
Conversions of all of these aspects could be done to keep them preserved, especially at an automated scale, but if society evolves to not update especially older documents, or comment systems, to update their compatibility with newer devices, that could lead to some specific subsections of posts on the internet to become difficult to preserve from an archaeology standpoint, and may lead to deteriorating conditions of the documents in question.
