There must be something magical about this number. People who are new to long range shooting ask about shooting at a 1000 yards and people who are new to how much long range shooting costs ask if they can do 1000 yards under $1000.
Youtube is full of videos that address this topic and I started getting some questions on the subject as well. I figured I should offer my two cents.
I am going to approach this a little bit differently than most people do. Usually everyone starts out with choosing a rifle and then works hard to squeeze everything else into the remaining budget. I will start out by ignoring the rifle. I am not terribly pre-occupied with squeezing into exactly $1000. I'll start by laying out a couple of options for everything but the rifle and then make some rifle recommendations. We will probably end up above $1k, but hopefully not by much.
My purpose here is really to lay out what I would do while trying to stay on a tight budget. Keep in mind that I went through this exercise more than once when I was just getting into this, so I speak from experience. This is sort of a "knowing what I know now" exercise.
General disclaimer: there are affiliate and non-affiliate inks below, as appropriate.
Naturally, let's start with optics.
There are two competing (sometimes complementary) schools of thought on long range shooting. Some prefer to dial everything. Some prefer to hold. Both approaches are viable and have their strengths and weaknesses. Finding a budget riflescope that has a long history of tracking correctly is not so simple. If you want to dial, this road leads us to SWFA Classic fixed power scopes with either 10x42 or 12x42 with Mil-Quad reticle being my recommendations. Either scope will cost you $300 if you can find one in stock. The idea, with either one of these, is to dial for elevation and hold for wind.
https://www.swfa.com/swfa-ss-10x42-tactical-30mm-riflescope-3.html
https://www.swfa.com/swfa-ss-12x42-tactical-30mm-riflescope-105770.html

If you prefer to not dial, you need something with a tree reticle that allows you to compensate for both drop and wind. In this case, tracking accuracy is not critical, but you want something that will stay zeroed and give you at least 10mrad of holdover in the reticle. This is not the time to chase after high magnification or something particularly feature rich. You want this scope to hold zero, and offer a decent FFP reticle. SwampFox Patriot 4-16x44 with Sharpshooter Mil reticle for $300 is a good example of such a scope:
https://bit.ly/3nD89vg
Another is the likely related Vortex Diamondback Tactical 4-16x44 with EBR-2C mrad reticle for $350: https://bit.ly/3biLzlK
Neither is going to get you to give up your S&B, but they have been around for a bit, generally hold up well and have very competent reticles.
Now, let's talk rings and bases. The cheapest way to get to 1k is with a bolt gun, so I am assuming a conventional bolt action rifle that will require a picatinny base and two rings. This is not where you want to skimp out. I have seen a ton of problems when people thought that a set of $20 pot metal rings is fine because some idiot at a gun show told them "it is all the same". It is not.
My general recommendation is to get a set of Burris XTR Signature rings of appropriate diameter and height, which, for the scopes above is probably going to 30mm and low or medium height for around $100.
https://bit.ly/3vVnR8G
The inserts in the Signature rings give you a lot of latitude in correcting some of the rifle machining issues and in dialing in appropriate slope.
If you would rather have something simpler and less expensive, I have been pleasantly surprised by the UTG Pro rings. They are made in the US and have been quite consistent: https://bit.ly/3jMIi2V
As far as the Picatinny rail goes, this sorta depends on which action it is for. For example for Tikka T3x, I think Area 419 (https://bit.ly/3nxHGPC) is a good budget option for $60. Generally, I have had good luck with EGW, Badger and several others. Go with the 15 or 20 MOA version.
That more or less completes the optics part and we have spent, depend on specific choices between $420 and $510. That's essentially half of the $1k target budget, so I think we will overshoot it by quite a bit.
Assuming you are planning to shoot either off the bench or prone, you will need both front and rear supports. Shooting prone off of a bipod is a valuable skill, so let's assume you need a bipod and a small rear bag of some sort.
There are many expensive and excellent bipods out there and we will ignore just about all of them. A top end triple pull Ckye bipod alone can cost you $1k, so it is well outside our scope here (although I plan to pick one up soon because I am basically a gun snob; https://bit.ly/3vUuurX). For the longest time, Harris bipod for around $100-$140 was the default choice and I would probably lean that way (https://bit.ly/3vSXv7c), but there is a bunch of various Harris clones out there marketed by Blackhawk, Caldwell, Champion, etc that cost less. Their QC is occasionally shaky, but given what we are looking to do here, one of the clones would probably work fine. For me, 9-13" height pivoting models seem to work best: https://bit.ly/2ZIK0v7
For a small rear bag that you can squeeze to make small POA adjustments, something like this Armageddon Gear $35 product will work well: https://bit.ly/31c7w4e
I mostly use a similar one from Eberlestock, but they work the same way.
We are now somewhere between $550 and $700 in and we have not even started with the rifle itself.
Let's first think about the caliber.
In principle, 308Win is the old standby, but I would lean toward 6.5Creedmoor or 243 Winchester. They shoot flatter and have less recoil. If you plan to also hunt with this gun, stick with the 6.5Creedmoor.
If you are on a budget, a dedicated heavy barrel precision rifle is likely not in the cards unless you find a really good deal on a used gun. Something that looks like a conventional hunting rifle might be your best bet, but given the rather thin barrels on these guns, make sure you wait long enough between shots for the barrel to cool down.
There are plenty of inexpensive rifles from Savage (Axis) and Ruger (American) and I have seen some really accurate ones and some really inaccurate ones. For consistency, I lean toward Tikkas, though finding a new one for less than around $700 is kinda rough, but they exist: https://bit.ly/3pN2wNv
I would probably try to find a used CTR or a new one in an unpopular caliber if you are a reloader: https://bit.ly/3jLRWmh
If you take your time, there are deals out there. A while back, when going through this exact exercise, I managed to pick up a Tikka 695 in 280Rem for $500. I configured it at first with a 10x SWFA scope in a manner very similar to what I described above. It cost me right around $1k overall. 280Rem is never really looked at as anything but a hunting caliber, but I loaded it up with 150gr SMKs because I wanted to do some target shooting. The rifle turned out to be stupid accurate (I still have it, but I restocked it since) and that 150gr load made 1000 yards not that challenging unless the wind was doing something weird. I did spend a lot of time practicing though.
What did we learn through this whole exercise?
Once you have the rifle, setting it up with the right optics, etc will cost you at least $500. If you really want to squeeze under $1k, you need a sub-$500 rifle that is accurate enough. They do exist, but you may need to have patience and look for deals. Make sure the stock allows the barrel to be truly free floated.
Lastly, what is accurate enough? Let's assume you are looking to hit an 18" plate at 1000 yards. If you rifle prints 1MOA groups, that means that the remaining margin for you is ~0.5MOA or ~0.15mrad. That is by how much you can misread wind and mishandle the rifle and still hit that 18" plate.
If you are new at this, get some professional instruction.
If you are not new at this, continue getting some professional instruction.