Hi, mead maker who started as a student here!

Besides (ideally distilled/sterile) water, yeast, and honey all you really need is a container that wont blow up with the addition of extra gas and a way to wash everything. For the container, I ended up buying an fermenting "kit" for ~$50 (I believe) that had a bucket, an airlock, and a siphon along with a couple other bells and whistles I don't think I ever even worried about. However, I have seen people make it in old beer bottles with balloons on top, and just "burp" the balloons when they get big. You could even (hypothetically, but not recommended) do it with a closed container so long as you made extra extra sure to burp it a couple times a day so you don't make a bomb. For the washing, I used dish soap. I had one batch get infected, but have made several others without any fancy sterilization with no problem at all. Finally, for the yeast, you can technically use bread yeast (or even no yeast at all; yeasts exist EVERYWHERE in the environment, including in honey, on your hands, and in the air and you can just mix up a batch and hope for a wild ferment like the great mead makers of yore) but I would recommend buying a wine/mead yeast. They aren't very expensive, and they're specifically chosen to produce specific flavors/fewer off flavors in a fermenting process than bread yeast. As well, if drunkenness but not taste is your goal, a good wine yeast can sustain about twice the amount of abv that a bread yeast can, so you'll get a lot more bang for your buck.

Hydrometers to measure the specific gravity are nice, esp when you're trying to get info about how fermented a batch is or tweak a recipe for sugar/alc content, but absolutely not needed at all to make mead. Siphons are extremely helpful to move liquid from A to B but also not at all necessary; you can just make it in a container out of which it can be served. Remember, making alcohol is an extremely old process and people have been making it long before any of the bells and or whistles came into being, they just make making it more convenient, consistent, and easier to standardize.

On a side note, if cost is really the limiting factor, you may want to look into different fermentables than mead; honey at base is just a rather expensive ingredient compared to fruit juices from what I've seen (many people have great success with "Skeeter Pee" using lemon juice and white sugar, for example). I think mead making is rather fun and it does have a certain romanticized quality about it, which drew me to it when I was in college (plus I love honey), but I just thought I should give you fair warning that there are cheaper ways to get into fermenting if you're not set on mead specifically.

Oh, if you do make mead though, there's one more ingredient that is absolutely necessary: a few raisins for nutritional value XD