Selfhosted

I use Jellyfin as a home media server - in my set up I have it running on my desktop PC, and I use it to stream a media library to my tv.

A home media server basically just means its meant to be deployed at a small scale rather than as a platform for 1000s of people to use.

Your scenario is exactly what Jellyfin and Plex can do. If you have 5 users then you just need a host device running the server that is powerful enough to run 5 video streams at the same time. The server can transcode (where the server takes on the heavy lifting needing a more powerful CPU) or direct play (where all the server does is send the bits of the file and the end user's device such as a phone or smart tv does the hard work of making a quality play, so a lower power server device can work).

If this is contained within your home, your home wifi or network should be fine to do this, even up to 4k if your network is good enough quality. If the 5 people are outside your home then your internet bandwidth - particularly your upload bandwidth - and your and their internet quality will be important determinant of quality of experience. It will also need more configuring but it is doable.

This doesn't need to be expensive. A raspberry pi with storage attached would be able to run Jellyfin or Plex, and would offer a decent experience over a home network if you direct play (I.e. just serve up the files for the end users device to play). You might need something more powerful for 5 simultaneous direct play streams but it would still be in the realms of low powered cheap ARM devices.

If you want to use transcoding and hardware acceleration you'd need better hardware for 5 people to stream simultaneously. For example an intel or amd cpu, and ideally even something with a discrete graphics card. That doesn't mean a full desktop PC - it could be an old PC or a minipc.

However most end user devices such as TVs, PCs, Phones and tablets are perfectly capable of direct playing 1080p video themselves without the server transcoding. Transcoding has lots of uses - you can change the audio or video format on the fly, or enable streaming of 4k video from a powerful device to a less powerful device - but its not essential.

Direct play is fine for most uses. The only limitation is the files on the server need to be in a format that can be played on the users device. So you may need to stick to mainstream codecs and containers; things like mp4 files and h.264/avc. You could get issues with users not being able to playback files if you have say mkv files and h. 265/hevc or vp9. Then you'd either need to install the codecs in the users device (which may not be possible in a smart tv for example) or use transcoding (so the server converts the format on the fly to something the users device can use but then needing a more powerful server)

I prefer Jellyfin as its free and open source. It has free apps for the end user for many devices including smart tvs, streaming sticks, phones, tablets and PCs. Its slightly less user friendly than plex to set up but not much. And the big benefit is your users are only exposed to what you have in your library.

Plex is slightly more user friendly but commerical. You have to pay for a licence to get the best features and even then it pushes advertising and tries to get your users to buy commercial content. Jellyfin does not do that at all.

Finally if your plan is to self host in the cloud, again this is doable but then you stray into needing to pay for a powerful enough remote computer/server, the bandwidth for all content to be served up (in addition to your existing home internet) and the potential risk of issues with privacy and even copyright infringement issues around the content you are serving. A self hosted device in your home is much more secure and private. A cloud hosted solution can be secure but youre always at risk of the host company snooping your data or having to enforce copyright laws.

Edit: the other thing to consider ia an FTP server. If you just want to share the files, its very simple to set up. What Jellyfin and Plex offer is convenience by having a nice library to organise things, and serving up the media. But direct play from a media server is not far off just downloading the file from an ftp server to your home device and playing it. But you can also download files from a Jellyfin server so I'd say its worth going the extra step and to use a dedicated media server over ftp.