As others have said, pretty much any DSLR (or interchangeable lens mirrorless camera) will do the trick and its more about lenses and lighting. Nikon and Canon DSLRs have the biggest range of lenses available, especially if you want to get them cheap second hand.
A proper macro lens (or called 'micro' for Nikon lenses) is by far the easiest/most versatile option and you can often pick them up for $2-300 second hand. Cheaper than that a 35mm or 50mm lens (older one with mechanical aperture ring) on a reverse adaptor can give good results but is trickier to use (full manual control and you focus by moving the camera), or you can use a regular lens with extension tubes. You can also combine a macro lens or reversed lens with extension tubes for more magnification, or get in deep and do weird things like using adaptors to reverse mount a 50mm lens onto the front of a big telephoto lens.
For lighting, an external flash is best - you need to use a small aperture to get enough depth of field up close, which means you need lots of light. On-board flashes can be ok if you diffuse it with a plastic bag or something, but tend to give quite a harsh direct light up that close. I tried a cheap LED ring flash but it didn't put out enough light to really be useful.
I have an album of macro stuff here: https://flic.kr/s/aHsjPKoEpi each image should have camera and lens info below it once you click on it so you can see what gear and settings I used.
Getting good at it takes patience and practice so don't give up if you aren't getting good results at first, keep trying and experimenting with different settings.
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