Our children just fill a small plastic bowl with water and take their paint to whatever table they want to use. We use all washable watercolor, and they keep the paints away from the upholstered furniture.
Sometimes you do have to buy organizing stuff like boxes, baskets, folders, etc. I sure wouldn’t want loose watercolors and brushes in my kitchen cabinets. You can get cheaper plastic boxes at Walmart and Target. They have lids and usually cost $1-2. Not bad at all, and you can fit a lot in them.
I find it much more worth my time and money to have art supplies accessible to the children, not just to me. It saves my sanity when I know they can reach and get whatever they need without needing me to rearrange an entire closet or cabinet for them to find what they want. Inevitably, they want that “thing” on the top shelf in the back, am I right?! The boxes I got at Hobby Lobby were originally $3.99, but they were on sale for 50% off. $2/box several years ago, and they’re still holding up well.