As I understand it the failure rate is the number of failures per unit time. I'm reading a book in which a collection of independent random variables $X_1, X_2, ...$ are assumed to represent the time until failure of the $i$th component, and are assumed to be exponentially distributed with parameter $\theta$. So $X_i$ has pdf
$$f(x)=\theta e^{-\theta x}$$
Since $E[X_i]=1/\theta$ then this represents the mean time until first failure, and $\theta$ represents the failure rate, the number of failures per unit time. However, the book writes that we assume there are about 0.5 failures per year, so I would think $\theta=0.5$. However, it goes on to say this is the same as a $\Gamma$-distribution with $\alpha=1$ and $\beta=2$. We define the $\Gamma$-distribution as
$$g(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$$
That makes me think $\theta=\beta=2$ ... so I'm now confused about what the meanings of these parameters are.
If it's useful, I'm reading Degroot and Schervish's Probability and Statistics Fourth Ed., chapter 7, example 7.1.1.
In this specific case, I think you are not understanding the example correctly. The book does not say that these two are the same distributions. It says that given $\theta$, the samples $X_i\sim\mathrm{Exp}(\theta)$ which has the pdf $f_X(\theta)=\theta e^{-\theta x}$ as you mentioned.
What is says next, is that there is a prior belief about the parameter $\theta,$ which is that $\theta$ is around $0.5$. An example of such a prior is the Gamma$(1,2).$ Indeed, this Gamma distribution has mean $1/2=0.5$ and hence supports the prior belief.