Does RBC models pin down equilibrium output, amount of labor and capital without initial condition?
The stock of capital in the RBC model is a predetermined state variable, meaning that the capital $K_t$ available at time $t$ was determined at time $t-1$. Usually we have an accumulation relation of the form 
$$K_t = I_{t-1}+(1-\delta)K_{t-1}$$
When solving an RBC model starting at time $t=0$, we must exogenously assume some initial capital stock $K_0&gt;0$ to get the model started, since we don't explicitly consider what happened at time $t=-1$.

Labor, on the other hand, generally is not specified ahead of time. Instead, in many RBC models the household has an exogenous certain time endowment, perhaps normalized to 1, and then "leisure" $L_t=1-N_t$ is the amount of time remaining after "labor" $N_t$. Leisure then enters into the utility function just like consumption. (An equivalent and common alternative is to directly put labor $N_t$ as a "bad" into the utility function.)

The basic contrast is that capital is a stock while labor is a flow. We can choose the level of investment flow $I_t$ in each period, and the stock $K_t$ (which we can no longer control once we've reached time $t$) reflects the accumulation of past investment flows $$K_t = \sum_{s=0}^\infty (1-\delta)^s I_{t-s-1}$$
Meanwhile, labor $N_t$ is a flow that the household can choose optimally at time $t$.

Things becomes more complicated in a model with either (A) human capital or (B) search frictions. With (A) human capital, the effective supply of labor is partly a stock, since it reflects past flow investments in human capital; whereas with (B), the number of ongoing employer-employee relationship is a stock that reflects accumulated matches. Neither of these are present in the baseline RBC model, however. 

Instead, the baseline RBC model is essentially the Ramsey growth model augmented with a stochastic process for TFP and endogenous labor supply. In this model, capital $K_t$ is the only endogenous linkage between periods.