I am watching a video on a star that is supposedly older than the universe. HD140283 (Methuselah). According to the first 20 years of research on the star, scientists believed the star is just over 14B years old based on the amount of oxygen in it. Scientists now acknowledge they made several errors in the study of the star. It is a second generation star with an age of about 11.5 - 12B years old.
My question is: How has it lasted so long? (It is a metal poor subgiant blue star).
It seems to me that you have not realised that stars live so long in general. For example, it seems likely that the Sun (which has a current age of 4.57 billion years) will not consume all its core hydrogen for another 5 billion years and will only reach the tip of the red giant branch (as a hydrogen shell-burning star) when it is 12.2 billion years old (e.g. see Schroeder & Smith 2008). So there is no problem in general with stars living as long as HD 140283. It is a star with a bit less than a solar mass, that would have been born very early in our Galaxy's history (perhaps a billion years after the big bang), and as a result has a very low "metal" (any element heavier than helium) content, since previous generations of stars would have been required to enrich the interstellar medium with such material.
But why do stars live so long? Well, partly it is to do with the amount of fuel available and partly with how slowly the fuel is "burned" in nuclear reactions. The Sun converts about 4.2 million tonnes of mass into energy every second (just divide the Sun's luminosity by $c^2$); but the Sun has a mass of $2\times 10^{27}$ tonnes. If you know that the hydrogen burning pp chain, that turns hydrogen into helium, has an efficiency of 0.7% (i.e. 0.7% of the mass is converted to energy), that all of the Sun's material is hydrogen (it isn't, but most of it is), and that all of it can be consumed in nuclear reactions (in reality only about 20% can in the Sun), then you see that Sun has enough fuel to burn at its current rate for about 10 billion years.
But why is the pp-chain so slow? The reason here is nuclear physics. The pp-chain involves getting two protons close enough together that they feel the strong nuclear force. The protons repel each other and so this step is quite difficult. It requires high temperatures and quantum tunneling to overcome this "Coulomb barrier". But even when that has happened, the di-proton" is not stable and will fall apart before fusion can take place. In order to provide a fuseable material, one of the protons has to change into a neutron, and the combination of a neutron+proton (known as a deuteron) is stable. The problem is, that this conversion of a proton to a neutron is moderated by the weak nuclear force and is extremely unlikely to take place. As a result, the average proton in the core of the Sun survives for about 10 billion years before fusing with another proton to form a deuteron. This weak interaction is the rate-controlling step of the whole chain.
Thus I would argue that stars live so long because the probability of a proton turning into a neutron whilst part of an unstable di-proton is very small.
EDIT: An interesting twist to all this is the particular star in question. As you say, it is a very metal-poor, but its surface temperature is about the same as the Sun's. The lifetimes of metal-poor stars are actually shorter than metal-rich stars of the same mass (e.g. Bazan & Mathews 1990), because a lower metallicity leads to lower opacity and allows radiation to escape more easily and they have higher luminosities. However, the lower metallicity leads to higher surface temperatures at the same mass, so although HD 140283 has a similar temperature to the Sun, it is significantly less massive ($0.8 M_{\odot}$ - Creevey et al. 2014).
The lower mass and the lower metallicity combine to give HD 140283 almost the same main sequence lifetime as the Sun. However, HD 140283 has now left the main sequence and is in the subgiant phase, with a radius of $\sim 2R_{\odot}$. It is this that enables a reasonably accurate age estimate of $\sim 13$ billion years.
The lifetimes of stars are basically set by the combination of how much fuel they have for nuclear fusion (roughly proportional to their mass) and how fast they consume that fuel (e.g., by fusing hydrogen into helium). The fuel consumption rate -- which determines the star's luminosity or absolute brightness L -- is also set by their mass, but in an exponential fashion: $L \propto M^{4}$ (for stars with masses $< 30$ times the mass of the Sun).
This means that the lower a star's mass, the longer it will live. A star with the mass of the Sun is predicted to have a lifetime of $\approx 10$ billion years. A star with a mass of 10 times the Sun will only life for a few tens of millions of years, while a star with 0.1 times the Sun will have a lifetime of trillions of years. So all that's required for a star like HD 140283 to "last this long" is to have a mass somewhat less than the Sun's; the comparison with stellar evolution models suggests it has a mass of about 0.8 times that of the Sun.
(Strictly speaking, these numbers apply to the "main sequence" lifetime of the star -- the period when it's fusing hydrogen in its core. But that is by far the longest part of a star's life, so it determines most of the answer.)
