Diamond wire bandsaw
Diamond wire is available in thin widths (0.1 – 0.6 mm), for $1/m.
Modify a bandsaw to use this wire and use with water cooling.
Here is a video documenting a project to add cooling to a bandsaw, Youtube.
The wire comes linear, the ends would need to be joined (brazed or butt welded). Welding steel requires black flux, like Harris Stay Silv Black Flux. Brazing steel can be done with with silver solder and standard flux. This video shows brazing bandsaw blades, youtube.
The wire wears out quickly, so a longer loop would be better.