Can a gas cloud of pure helium collapse and ignite into a star? Assuming there could be a giant gas cloud with negligible amount of hydrogen and metal (elements with atomic number $Z\geq3$), could it collapse gravitationally and form a pure helium star that would skip the main sequence entirely?
I think it's possible in principle but my main concern is that the ignition of $^4$He is a 3-body reaction (the triple alpha process) and requires a higher temperature and a higher density than the pp-chain or the CNO cycle. Would the gas keep collapsing until it reaches the critical density for carbon formation or would it reach hydrostatic equilibrium before that point, preventing further collapse and star formation?
Perhaps it's only a matter for the cloud to have a minimal mass that allows the nuclear reaction? If so, how can I predict this minimal mass?
 A: The answer is that if you ignore degeneracy pressure, then indeed a collapsing cloud of helium must eventually reach a temperature that is sufficient to initiate the triple alpha $(3\alpha)$ process. However, electron degeneracy pressure means that below a threshold mass, the centre will not become hot enough to ignite He before the star is supported against further collapse.
The virial theorem tells us that (roughly)
$$ \Omega = - 3 \int P\ dV\ ,$$
where $\Omega$ is the gravitational potential energy, $P$ is the pressure and the integral is over the volume of the star.
In general this can be tricky to evaluate, but if we (for a back of the envelope calculation) assume a uniform density (and pressure), then this transforms to
$$ -\frac{3GM^2}{5R} = -3 \frac{k_B T}{\mu m_u} \int \rho\ dV = -3 \frac{k_B T M}{\mu m_u}\ , \tag*{(1)}$$
where $M$ is the stellar mass, $R$ the stellar radius, and $\mu$ the number of mass units per particle ($=4/3$ for ionised He).
As a contracting He "protostar" radiates away gravitational potential energy, it will become hotter. From equation (1) we have
$$ T \simeq \left(\frac{G \mu m_u}{5k_B}\right) \left( \frac{M}{R} \right)$$ 
Thus for a star of a given mass, there will be a threshold radius at which the contacting protostar becomes hot enough to ignite He ($T_{3\alpha} \simeq 10^{8}$ K). This approximation ignores any density dependence, but this is justified since the triple alpha process goes as density squared, but temperature to the power of something like 40 (Clayton, Principles of Stellar Evolution and Nucleosynthesis, 1983, p.414). Putting in some numbers 
$$ R_{3\alpha} \simeq \left(\frac{G \mu m_u}{5k_B}\right) \left( \frac{M}{T_{3\alpha}} \right) = 0.06 \left(\frac{M}{M_{\odot}}\right) \left( \frac{T_{3\alpha}}{10^8\ {\rm K}}\right)^{-1}\ R_{\odot} \tag*{(2)}$$ 
The question then becomes, can the star contract to this sort of radius before degeneracy pressure steps in to support the star and prevent further contraction?
White dwarfs are supported by electron degeneracy pressure. A "normal" white dwarf is composed of carbon and oxygen, but pure helium white dwarfs do exist (as a result of mass transfer in binary systems). They have an almost identical mass-radius relationship and Chandrasekhar mass because the number of mass units per electron is the same for ionised carbon, oxygen or helium.
A 1 solar mass white dwarf governed by ideal degeneracy pressure has a radius of around $0.008 R_{\odot}$, comfortably below the back-of-the envelope threshold at which He burning would commence in a collapsing protostar. So we can conclude that a 1 solar mass ball of helium would ignite before electron degeneracy became important. The same would be true for higher mass protostars, but at lower masses there will come a point where electron degeneracy is reached prior to helium ignition. According to my back-of-the-envelope calculation that will be below around $0.3 M_{\odot}$ (where a white dwarf would have a radius of $0.02 R_{\odot}$), but an accurate stellar model would be needed to get the exact figure.
I note the discussion below the OP's question. We could do the same thing for a pure hydrogen protostar, where $\mu=0.5$ and the number of mass units per electron is 1. A hydrogen white dwarf is a factor of $\sim 3$ bigger at the same mass  (since $R \propto $ the number of mass units per electron to the power of -5/3  e.g. Shapiro & Teukolsky, Black Holes, White Dwarfs and Neutron Stars). But of course it is not the triple alpha reaction we are talking about now, but the pp-chain. Figuring out a temperature at which this is important is more difficult than for the $3\alpha$ reaction, but it is something like $5\times 10^{6}$ K. Putting this together with the new $\mu$, the leading numerical factor in equation (2) grows to 0.5. Thus my crude calculation of a threshold radius for pp hydrogen burning would intersect the hydrogen white dwarf mass-radius relationship at something like $0.15M_{\odot}$, which is quite near the accepted star/brown dwarf boundary of $0.08M_{\odot}$ (giving some confidence in the approach).
