Technically, it means an object of finite spatial extent surrounded by vacuum, and nothing else in the entire universe. Obviously that's an idealization. :-) But it makes the mathematical model tractable.

> wouldn't such an object continue heating up "infinitely"

Not necessarily; it might reach a stable state where it can't radiate any more energy. Cold white dwarfs, cold neutron stars, and cold planets are possible examples of such states. ("Cold" here basically means "at absolute zero", i.e., in the ground state for its configuration of particles.)

Reminds me of Fritz Zwicky's "spherical bastards". Although unrelated, my physics prof that taught thermo introduced spherical bastards as an aside after stating we'd basically assume all bodies are spherical when calculating thermo transfers. This prof was where I learned that Zwicky often referred to some colleagues as "spherical bastards", and when when questioned why, he responded: "because no which way you look at them, they're a bastard." I'm sad I can't remember the prof's name; he was quite a character, a short, squat old man. Just add a beard and appropriate clothing, he'd have played a good Santa Clause. He had a minor role in the Manhattan Project, and would often share stories. Lectures were always fun. One of the few classes I never skipped.

Yes, I know, but this in itself does not preclude energy being emitted by radiation. To preclude that, the object needs to be in a genuine ground state--no internal transitions possible that can reduce its total energy. That's a considerably stronger condition than T << µ.