How were unit labor costs estimated by Gali and Gertler (1999)?
In Galı́ &amp; Gertler (1999) the authors state that marginal costs is given by:
$$MC_t = \frac{S_t}{\alpha}$$ where $\alpha$ is the labor elasticity (from the Cobb-Douglas prod. function), and $S_t$ is the labor share of income (i.e. what portion of output GDP goes to labor). Next they actually get around estimating $\alpha$ by expressing everything in percentage changes so they actually use:
$$mc_t = \frac{MC_t - MC_{t-1}}{MC_{t-1}}= \frac{\frac{S_t}{\alpha} - \frac{S_{t-1}}{\alpha}}{\frac{S_{t-1}}{\alpha}}$$
where alpha cancels out so you are left with just  change in the labor share of income $\Delta S_t = s_t$.
They even state in the paper:

The data we use is quarterly for US over the period 1960:1}1997:4. We use the
(log) labor income share in the non-farm business sector for $s_t$

So in this case they just seem to have directly taken the labor share of income, which is plausible data for labor shares of income exist without any 'assembly' required (e.g. see this Fred dataset as an example).