There are two Boolean vectors, which contain 0 and 1 only. If I calculate the Pearson or Spearman correlation, are they meaningful or reasonable?
The Pearson and Spearman correlation are defined as long as you have some $0$s and some $1$s for both of two binary variables, say $y$ and $x$. It is easy to get a good qualitative idea of what they mean by thinking of a scatter plot of the two variables. Clearly, there are only four possibilities $(0,0), (0,1), (1, 0), (1,1)$ (so that jittering to shake identical points apart for visualization is a good idea). For example, in any situation where the two vectors are identical, then by definition $y = x$ and the correlation is necessarily $1$. Similarly, it is possible that $y = 1 -x$ and then the correlation is $-1$.
For this set-up, there is no scope for monotonic relations that are not linear. When taking ranks of $0$s and $1$s under the usual midrank convention the ranks are just a linear transformation of the original $0$s and $1$s and the Spearman correlation is necessarily identical to the Pearson correlation. Hence there is no reason to consider Spearman correlation separately here, or indeed at all.
Correlations arise naturally for some problems involving $0$s and $1$s, e.g. in the study of binary processes in time or space. On the whole, however, there will be better ways of thinking about such data, depending largely on the main motive for such a study. For example, the fact that correlations make much sense does not mean that linear regression is a good way to model a binary response. If one of the binary variables is a response, then most statistical people would start by considering a logit model.
There are specialised similarity metrics for binary vectors, such as:
For details, see here.