You can, but the results will be messy and sub-optimal compared to industrially produced cores.
I've done it for an inductor for which there wasn't a ready made core. It left me with something that sort-of worked but the grade was much lower than even the poorest commercial grade. To improve on that I would have had to invest a lot of money.
Here is an overview of the steps in commercial ferrite core production:
https://www.powerelectronicstalks.com/2018/11/ferrite-core-m...
Edit: Oh, and the way I did it: take a thermally resistant resin with low shrinkage on curing and load it with as much iron powder as it will take before it starts clumping, pour into mold, let it set and then press it out of the mold.
That was an interesting read. However it doesn't look like small scale DIY at a high quality level would be particularly expensive in terms of tooling? Only in terms of time and personal experience.
Ball mills are trivial to DIY - electric motor, belt drive, couple pieces of copper tubing, drill some holes in a wooden frame, use large metal coffee cans, throw some ball bearings in.
Calcination is also easy. 1000 C is not a difficult temperature to deal with. You can hit that using plain old nichrome wire. You will need to purchase a few ceramic bricks of course.
The only slightly tricky step I'm seeing is sintering. The nickle-zinc option mentioned in the article will be substantially easier (lower temperature, oxygen atmosphere) but the required temperature still pushes up against the limits of nichrome wire. Given that propane can get you to the vicinity of 1900 C and tanks of that are sold at the grocery store (at least where I live) I'm going to say this step is also cheap and easy to DIY.
The real issue is that you can't do this at home if you live in high density housing. You will need a shop or a small back yard.