What happens to a body, initially at 300K, kept in isolated space? Will it's temperature drop to 0k?

In practice, no. In theory, also no.

The Universe is filled with photons with an energy distribution corresponding to $2.73\,\mathrm{K}$, called the cosmic microwave background (CMB). Every $\mathrm{cm}^3$ of space holds around 400 of them, so each second one $\mathrm{cm}^2$ is hit by roughly one hundred billion of these photons. That means that if you place your "stable body" in an ever-so-isolated box, the box itself will never come below $2.73\,\mathrm{K}$, and neither will the body inside. It will asymptotically go towards thermal equilibrium at $T = 2.73\,\mathrm{K}$.

Even if you magically removed everything else in the Universe but the body$^\dagger$, $0\,\mathrm{K}$ can never be reached. The reason is that this would imply zero motion of the atoms of the body, which is forbidden by law.

In practice, it is actually possible to have temperatures lower than the 2.73 K of the CMB. When a gas expands, it cools, and if it expands faster than it can be heated by the CMB, it can temporarily reach lower temperatures. This is the case for the Boomerang Nebula, which has a temperature of 1 K. The nebula will probably be heated to the temperature of the CMB in around 10,000 years or so.

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$^\dagger$_{Or waited billions of years for the CMB to cool.}