It depends very much on how far away you’re looking! Most things out there could be considered “a far away distance”, even when we’re dealing with objects within our own solar system, but the times involved to travel between Mars and the Earth are much closer to nearly real-time than they are if you start venturing further afield.
Fundamentally, information can only travel through the Universe at the speed of light, and the larger your distances get, the longer it takes light to cross those distances. For anything happening on the Earth, this is not usually much of an impediment, because the distances involved in circling the Earth are not so great. To get from the surface of the Earth to the ISS (a distance of 408,000 meters), light, which travels at 299,792,458 meters every second, is only going to spend about a thousandth of a second (0.0013 seconds) in transit. Events on the ISS can therefore be considered pretty much real time, even though there is a measurable communications lag.
If you go further afield, but still within our solar system, light takes about 1.25 seconds to get to the Moon (so a two and a half second round trip), eight minutes to get from the Sun to the Earth, or about twelve and a half minutes to get to Mars. This all starts to build up to a more considerable time delay, but these are manageable delays - if I got an email response from someone I was writing to in less than 24 minutes I’d think that was pretty rapid.
Once you try talking to the outer solar system, the time delays get a little more significant. The light travel delay to New Horizons when it was swinging past Pluto was about four and a half hours, so to ping New Horizons and hear back instantaneously from the craft, you’d be waiting about nine hours. Somewhere around this kind of time delay, we might start to classify things as happening “in the past”, but this is still a time delay on functional human timescales. Nine hour delays are sending an email to someone and hearing back in the morning. Not so convenient, especially if something complicated is happening in that time, but also not the worst.
It’s when we start looking beyond our solar system and into the Milky Way as a whole, or towards other galaxies that the time delay, which has just been scaling up with the distances involved, gets a little more outrageous. To get information from the center of our own galaxy out to Earth, you have to wait over 26 thousand years. That is no longer a length of time I can wait for an email reply. Information that reaches the Earth from the center of our galaxy is as up to date as it can be, but it’s reporting on changes that happened 26,000 years prior. The changes we see, therefore, are happening at whatever speed we see them happening, but with a time-lag. If we teleported there, it’d be old news.
You can imagine that the further out we go, the bigger this problem gets. So, scrolling outwards, the next big thing is Andromeda, which is so far from us that light has been stretching towards us from those stars for 2.5 million years. I think by most standards, this would be considered observing the past, and yet it’s the closest (and therefore informationally least out of date) galaxy we can look at! Most of the rest of the galaxies in the Universe are much further away, and therefore any changes that happen within them are going to be reported to us by our cosmic messenger in light many millions or billions of years later. The one above is 100 times further away than Andromeda, so news from that galaxy will take 100 times longer to reach us.
Where exactly you feel you should put the boundary between “pretty close to real-time” and “definitely looking at the past” is a bit of an arbitrary, fuzzy boundary. If you want to use “how long would you wait for an email reply” as your metric (as I have here), then your boundary is somewhere within the confines of the solar system. But no matter what you want to put down, there comes a point where we are definitely looking into the past, and certainly by the time we’re looking at other galaxies, we’ve reached it.