Start brewing for as little as $10.
TLDR: This is technically the only equipment you need. https://www.amazon.com/Home-Brew-Ohio-Fermenter-Stopper/dp/B014T3LHFA/.
The first question anyone should ask themselves before starting any brew is do I have the right equipment. In general wine and mead brewing equipment is interchangeable. A carboy or a food grade bucket and some basic bottling equipment and/or sanitizing equipment. Usually something like this: https://www.amazon.com/Premium-Wine-Making-Equipment-Kit/.
The issue I have with a kit like the one above is the size. Great for beer where a 5 gallon brew will cost anywhere from $40 to $90 to brew which is approximately 50 beers when you figure for waste it’s about $0.80 to $1.80 a 12oz bottle. Where a 5 gallon batch of mead can be between $100 to $200 which figuring in waste is about $2.00 to $4.00 a 12oz bottle.
Though it’s not a lot of money when you think about the $5.00 to $10.00 pint you get at your local bar. The real difference comes from time. Most batches of beer are ready to drink in 4 weeks. While most batches of mead are ready to drink in 6 months.
The last major factor to consider it the time it ties up your equipment, with 5 gallons of beer it will need to use your carboy for 2 week on average then it can be moved to bottles. Mead needs at least a month if not 2 months of fermentation time before you move it to bottles. Though personally I leave it in a fermenter for the full 6 months. No matter how long you decide to let it ferment you can’t use that equipment until it’s complete.
What I’m leading up to is that 6 months is a long time to wait to find out you made a bad batch and then you have 5 gallons of waste. It would be quite annoying to realize you’ve waste all that time and money. Which is why I use multiple one gallon jugs, a setup similar to the one below found at: https://www.amazon.com/dp/B005CIBAE0/.
If you really just want to get started without all the bells and whistles the $10 setup all you really need a jug with an airlock https://www.amazon.com/Home-Brew-Ohio-Fermenter-Stopper/dp/B014T3LHFA/ and some bleach to disinfect your equipment.
Making a batch one gallon at a time, then staggering a new batch by a week or more will give you the ability to hone your process as well as less of an issues with bad batches as. You can also start experimenting with fruits and spices. For this though you will want several 1 gallon jugs like this 4 pack: https://www.amazon.com/gp/product/B00D23OO90. You will also want the same number of airlocks and #6 stoppers as jugs so you don’t have to wait for an airlock to be freed up. I have a set of these I’m happy with this 3 pack: https://www.amazon.com/gp/product/B00IGY9P2G and this 5 pack https://www.amazon.com/Three-Piece-Airlock-Drilled-Fermentation-Kombucha/dp/B075DJFLBK/.
In conclusion you can start making your own mead for as little as $10 with a jug and an airlock or can go all out with a full kit. The only wrong choice is to not brew it at all.