I am looking to find some resources on public streaming of astronomical data. Does such a thing exist? In my explorations so far, I came across lots of nice answers linking to data in archives, but no resources that offer to stream astronomical data live to a client. Does anyone here have some resources to share in terms of this?
Live data from Earth orbit may work somewhat, live data from elsewhere are possible only intermittently and have comparably low bandwidth. Let's look at a few prominent examples and in the end a few further issues concerning data distribution / availability.
ISS / earth orbit
The newest addition of the ISS in its low Earth orbit of about 400km is a broadband connection for not-live download of 50MBit. The bandwidth available for scientists with onboard experiments for live data is a shared bandwidth of around 4Mbit (the milage may vary a bit where exactly an experiment is located and how it is connected - but that's the best we get offered).
You may get an idea of the typical bandwidth available when you look at your weather forcast and the cloud or rain prediction and history. That's likely as much images as you can get from geo-stationary orbit for reasonable prices.
One of the best near-live data available is from the SOHO, the solar heliospheric observatory. It sits at the langrange point towards the Sun and has time in the deep space network (DSN), that is ground stations around the world, which can receive its communication: https://soho.nascom.nasa.gov/data/realtime-images.html - Mind that also this is not anything movie-like, but an image every so often. It needs several receiving antenna as the Earth rotates - and you always need an antenna capable of receiving the signal so that you don't loose information; that means you need at least 3 antenna for continuous coverage. These are large telescopes, it is expensive to build and operate - so scaling that is not easy and cheap.
If you look at the bandwidth available from solar system space craft, it looks very different, and different intstruments send their data consecutively when there is available a link.
I will mention two examples here: The Rosetta spacecraft had two channels (S- and X-band) which allowed communication at speeds between 8bit/s up to 91000bit/s (yes, bit/s, not kbit or MBit). ESA writes itself:
During the mission, the rate at which data could be sent from Rosetta to Earth varied from 10 to 22 000 bits per second. However, the rotation of the Earth meant that real-time communications were not always possible. The spacecraft was visible from the New Norcia antenna for an average of 12 hours per day. In addition, there were several periods of communications black-out when the spacecraft passed behind the Sun. To overcome these breaks in communication, Rosetta’s solid-state memory of 25 Gbits capacity was able to store all scientific data and then transmit them to Earth at the next opportunity.
The Mars rover perseverance seems to have slightly higher bandwidth, but with 800 bit/s up to 2000000bit/s it doesn't allow exactly live feeds of images either - especially as you have regular loss of communication because there is no line of sight due to the rotation of Mars (and partially also Earth). Thus the live data from it are really summed-up here on its overview page and you can filter by week. This week (2-8 April 2023) it lists around 2000 images with the best shown on the "image of the week" webpage.
Thus special schemes and plans and protocols are devised when which data are recorded and when they are being transmitted in which resolution. The data formats often need conversion, calibration and manual vetting to make sure that they can be understood and be interpreted meaningfully. E.g. for a magnetometer you need to know which other instruments were operating and the relative alignement in order to get meaningful magnetic data which does not show the magnetic field of the spacecraft and its electronics itself. For particle detectors and analysers, the raw voltages or charge numbers per channel are not directly useful either - but it needs calibration data which might depend on the settings for their magnetic and electrical fields etc - and you also need to know space craft orientation and where your sensor is, if you want to know where these particles come from.
Another stumbling stone is actually the security of these systems: all communication to spacecraft is made from separate networks which are not immediately accessible from the WWW. Setting up bridges poses a potential security risk - something which needs special care and active maintenance, more than a simple data dump needs where data are simply copied to from time to time - and thus in the end it needs money for people. That is done for the major missions but often that money and personel is not available for smaller projects and these tasks are done by people manually from time to time.
All this said, the major space organisations have a policy that they require that data are made available to the public - but it's not a requirement for live data. Instead the data are pre-processed and calibrated so that any interested 3rd-party who do not run the instruments (thus you or whoever, scientist or not) can actually understand and make use of the data without reading the complete technical hardware guide and software protocols of the spacecraft in question.
And last but not least: it's also our moral obligation to make these data available as we work with tax payer money. But it needs to be reasonably understandable data. Understandable is relatively easy for camera images - but it gets much more complicated for other instruments which often need some kind of manual calibration.
The bandwidth often is not HUGE, so as live as you can get is often already available; it is a few images per day at most.
Maybe you'd be interested in something like this: https://www.itelescope.net/
They let you rent time on well spec'd telescopes installed in dark sky sites, and you can point them and stream the data in real time.
Point a webcam at the sky. Technically that's a live stream of astronomical data.
(That's not just a joke. Some folks like meteor watchers really do that for actual sky observation.)
But seriously, the problem with your question is that "astronomical data" is a really broad category. You say in the comments that you're specifically interested in "radiation data emitted from astronomical bodies", but that really doesn't narrow it down much either, because almost all astronomy is based on observing (mostly electromagnetic) radiation from various bodies.
Also, a lot of astronomical data is only really useful or interesting after it has been analyzed, and it's often neither practical nor necessary to do the analysis in real time, so a live stream of the raw data often wouldn't be very useful (or interesting, even for just curiosity's sake — often the raw data is 99.9999% full of nothing interesting at all, with interesting things only appearing briefly and rarely at random).
Perhaps the most common exception to the above I can think of are various "space weather" applications where getting (relatively) frequently updated observations and analyses is useful for monitoring and forecasting. There are web sites like SpaceWeatherLive that provide aggregated access to things such as (near) real time solar activity data, including images from sun observation satellites, but you can also get some of the same data directly e.g. from NASA. Maybe that's something like what you're interesting in?