What is space temperature around Earth? What is the equilibrium temperature that a black body will reach at the Earth's distance from Sun?
Assume you have a spherical blackbody.

The solar flux at the radius of the Earth is given to a good approximation by $L/4\pi d^2$, where $d = 1$ au. This is $f=1367.5$ W/m$^2$ (though note the distance between the Earth and the Sun has an average of 1 au).

If it is a blackbody sphere it absorbs all radiation incident upon it. Assuming this is just the radiation from the Sun (starlight being negligible), then an easy bit of integration in spherical polar coordinates tells us that the body absorbs $\pi r^2 f$ W, where $r$ is its radius.

If it is then able to reach thermal equilibrium and it entire surface is at the same temperature, then it will re-radiate all this absorbed power. Hence
$$\pi r^2 f = 4 \pi r^2 \sigma T^4,$$
where $T$ is the "blackbody equilibrium temperature". Hence
$$ T = \left( \frac{f}{4\sigma}\right)^{1/4} = 278.6\ K$$