Neglecting the fact that this ball of gas would just collapse on itself - I'm curious if there's an agreed-upon measurement that takes into account the volume of all the ~100 billion stars to predict the diameter of a milky way where all the gas is in one orb. If not, is it safe to just use the volume of our Sun (which is an "average" star) and multiply by 100 billion or is there a more accurate data set to include in the calculation?
Asking for a 6-year-old :)
Obviously the answer depends on what the density of such a sphere was!
The only sensible way I can interpret this question is to ask what is the combined volume occupied by all of the galaxy's stars and then get a spherical radius from this.
A modal star in the galaxy has a mass of around 0.3 times the Sun and a radius of 0.3 times the Sun. Since the stellar mass distribution is quite peaked at that value, we could do worse than simply assume that we have about $10^{11}$ stars like this, since stars of higher and lower masses are less common.
In which case, the radius containing a similar total volume is just $0.3\times 10^{11/3}\ R_\odot= 1400R\odot$.
The problem with this calculation is that although the masses of stars are quite peaked, their radii are not. Some evolved stars individually have radii that are a significant fraction of the combined radius figure. Thus although the figure above might be good for the combined radius of all main sequence stars it certainly isn't a good number if you wanted to include all giant stars, with the densities they have now.
There is some simple back-of-the envelope reverse-engineering which might be worth sharing:
The following Wikipedia quote gives an approximate of the overall mass of the Milky way:
In March 2019, astronomers reported that the mass of the Milky Way galaxy is 1.5 trillion solar masses within a radius of about 129,000 light-years, over twice as much as was determined in earlier studies, and suggesting that about 90% of the mass of the galaxy is dark matter.
This would give around $1.5 \cdot 10^{11} M_\odot$ for the overall visible mass of the Milky way.
Assuming around 100 billion stars for the Milky way, we would end up with $1.5 M_\odot$ for an average star, if we assumed that all visible matter are stars. As PM 2Ring pointed out in a comment, this assumption is far from being correct.
In a nutshell, in terms of mass (not volume), approximating all stars with a single solar mass per star is at least roughly the same order of magnitude, which I found rather suprising.
Converting mass to volume via density, is not too straight forward though, because we would have to assume a size and a density (distribution) for all of the $n \approx 10^{11}$ stars.
