How small could our Sun be and still "burn" with nuclear fusion and emit the same spectrum of light and other radiation as the real Sun does?

Edit:

The goal is to have a small sun inside a huge vessel, such as an O'Neill cylinder.

Black body radiation

The Sun is, approximately, a black body. That means that the light it emits follows a particular spectrum according to Planck's law, with the shape of the spectrum determined solely by the Sun's surface temperature. In particular, the wavelength of peak emission can be found through Wien's law, which is also a function of temperature. Therefore, if we want our new star to emit light just like the Sun does, we need to keep it at the same temperature as the real Sun - about 5800 K.

On the main sequence, there are some simple scaling relationships between mass, radius, temperature and luminosity:
$$Lpropto M^3,quad Rpropto M^4/7,quad Tpropto M^4/7$$
assuming the proton-proton chain reaction is the main source of energy, which is the case. If we want to keep the temperature constant, we need to keep the mass constant, and thus keep the radius constant. Therefore, no main sequence star can be significantly smaller than the Sun and still have the same temperature.

The composition of the star does matter; in fact, according to something called the Vogt-Russell theorem, the mass and composition of a star uniquely determine its properties and evolution. This means that the exact form of the relations above does vary between stellar populations - in part, relying on the mean molecular mass $mu$ - but this still will not make a significant difference here.

Options for a Sun-like star

We could look at subdwarfs, low-metallicity stars that are dimmer than main sequence stars. They're not common, but they exist, lying immediately below the main sequence on the Hertzsprung-Russell diagram. All other G-type stars are either on the main sequence, and thus fairly like the Sun in size, or off the main sequence, as giants or supergiants. Therefore, our options are either G-type subdwarfs or main sequence stars that are slightly cooler than the Sun:

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  Tau Ceti, a main sequence star, might be a good choice; it has a temperature a few hundred Kelvin less than the Sun, and is approximately $0.8R_odot$ in size (Texeira et al. 2009).


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  Mu Cassiopeiae A, a subdwarf, has a similarly slightly-cooler temperature and again a radius of about $0.8R_odot$ (Boyaijan et al. 2008).

Another star I considered is Groombridge 1830. Like Mu Cassiopeiae, it's a borderline G-type subdwarf, but is about 1000 K cooler - too far from the Sun in terms of temperature.

Low-mass stars and true lower limits

You won't find a star small enough to fit into an O'Neill cylinder - at least, not one that formed by natural means. EBLM J0555-57Ab, a small, late-type red dwarf has a radius 0.85 times that of Jupiter (von Boetticher et al. 2017) - likely too large for your purposes. Bodies smaller than that are likely too low-mass to fuse hydrogen, and would instead be brown dwarfs. Of course, EBLM J0555-57Ab is also likely cool, with a surface temperature far below that of the Sun.

Brown dwarfs - many of which fuse deuterium - are not true stars, and are usually cool, with typical temperatures around 1000 K, producing spectra much different from the Sun's. Some exoplanets may be much hotter than this, with surface temperatures comparable to those of many stars (see e.g. Kepler-70b, with a surface temperature of about 7000 K as per Charpinet et al. 2011). However, those bodies are only hot because they're irradiated by the stars they orbit; on their own, they would not generate that much heat.

White dwarfs

There is a possibility I had completely forgotten about before: a white dwarf. Many white dwarfs are hot, with temperatures up to about 100,000 K or so. However, they do cool - albeit slowly, as they have small surface areas. This cooling takes a long time, but some white dwarfs have become cooler than the Sun. WD 0346+246 is a famous case, with a surface temperature of about 3900 K (Hambley et al. 1997).

This implies that white dwarfs with temperatures like that of the Sun do exist; moreover, they're small. The same group measured WD 0346+246 to have a radius roughly that of the Earth, which is extraordinary - certainly less than that of EBLM J0555-57Ab. The problem, of course, is that white dwarfs don't undergo fusion. Indeed, the degeneracy of the matter inside a white dwarf means that fusion reactions are unstable, and can lead to novae and Type Ia supernovae.

A note on spectral lines

Finally, I should mention that actual stellar spectra are extremely complicated. A black body spectrum only models the global behavior; on smaller scales, there are numerous absorption and emission lines. I wrote in a related answer that Dyson spheres and brown dwarfs could be distinguishable by the presence and absence of various spectral lines.

The same should be true here. Just because another body of some sort is the same temperature as the Sun doesn't mean it would appear identical to the Sun, under spectroscopic scrutiny. I assume you don't care much about this, but it definitely is something to be aware of. Atmospheric composition really does matter.

The mass of a star is directly related to how hot its surface is, which in turn, is responsible for the wavelengths of light it emits (This is called Black-Body Radiation).

As a main sequence G2V star, the sun has a surface temperature of 5778 K. A smaller main sequence star will be cooler and therefore redder. A larger star will be hotter, and therefore whiter.

The only stars that will emit the same wavelengths as the sun are those that have the same mass.

https://en.wikipedia.org/wiki/Main_sequence

https://en.wikipedia.org/wiki/Black-body_radiation