Let \(\mathcal A\) be a space of functions in the upper half space \(\mathbb R_{+}^{d+1}=\{(x,y):x\in \mathbb R^{d}\), \(y>0\}\) or in the unit disc \(\mathbb D\) of the complex plane. Let \(\mathcal C\subset\mathcal A\) be the set of continuous functions and let \(\tau\) be a topology in \(\mathcal A\) such that \(\mathcal C\) is dense in \(\mathcal A\). A relatively compact subset \(F\subset\mathbb R_{+}^{d+1}\) (or \(F\subset \mathbb D\)) is called a Farrell set (for \((\mathcal A, \tau)\)) if for any \(f\in \mathcal A\) bounded on \(F\) there exists a sequence of continuous functions \(\{p_n\}\) in \(\mathcal C\) tending to \(f\) (in the topology \(\tau\)) and pointwise-boundedly on \(F\), that is, \(p_n\to f\) pointwise on \(F\) and \[ ||p_n||_F\to ||f||_F, \] where \(||f||_F\) denotes the supremum of \(f\) over \(F\). Given a point \(x\in\mathbb R^d\) and \(\alpha>0\), let \[ \Gamma(x,\alpha)=\{(y,t)\in\mathbb R_{+}^{d+1}:|y-x|\leq\alpha t\} \] be the cone with vertex at \(x\) and aperture \(\alpha\). Given a set \(F\) in the upper half space \(\mathbb R_{+}^{d+1}\), the point \(x\in\mathbb R^d\) is in the nontangential closure of \(F\), if there exists \(\alpha>0\) such that \(x\) is in the closure of \(F\cap\Gamma(x,\alpha)\). The nontangential closure of \(F\) is denoted by \(F_{nt}\). In the Euclidean space \(\mathbb R^d\), let \(Q\) denote any cube with sides parallel to the axis and write \(|Q|\) for its Lebesgue measure. A locally integrable function \(f\) on \(\mathbb R^d\) has bounded mean oscillation, \(f\in\text{BMO}(\mathbb R^d)\), if \[ ||f||_{*}=\sup_{Q}\frac{1}{|Q|}\int_{Q}|f-f_Q|<\infty, \] where \(f_Q=\frac{1}{|Q|}\int_{Q}f\) is the mean of \(f\) over \(Q\). The space of functions of vanishing mean oscillation \(\text{VMO}(\mathbb R^d)\) is the closure in BMO of the continuous functions with compact support. Given a function \(f\) in \(\text{BMO}(\mathbb R^d)\), \(f(z)\) denotes the value of its harmonic extension at the point \(z\in\mathbb R_{+}^{d+1}\). The space of harmonic extensions of functions in \(\text{BMO}(\mathbb R^d)\) is denoted by \(\text{BMO}(\mathbb R_{+}^{d+1})\). In this paper the authors study Farrell sets for the space of functions of bounded mean oscillation obtaining a geometric description as is shown in Theorem 1 and its analytic counterpart. They prove the following main theorems: Theorem 1. Let \(F\) be a relatively compact set in \(\mathbb R_{+}^{d+1}\). Then the following conditions are equivalent: (a) \(F\) is a Farrell set for \(\text{BMO}(\mathbb R_{+}^{d+1})\) equipped with the weak-\(\ast\) topology. (b) \(F\) is a Farrell set for \(\text{VMO}(\mathbb R_{+}^{d+1})\) equipped with the norm topology. (c) Almost every point of \(\bar{F}\cap\mathbb R^d\) is the nontangential limit of points of \(F\), that is, \(|\bar{F}\cap\mathbb R^d\setminus F_{nt}|=0\). The corresponding result is proved for BMOA and VMOA in case of the unit disk \(\mathbb D\). A set \(F\subset\mathbb R_{+}^{d+1}\) is dominating for BMO (for VMO) in its closure if for any \(u\in\text{BMO}\) \((u\in\text{VMO})\) such that \(\sup_{F}|u|<\infty\), one has \(u\in L^{\infty}(\bar{F})\). The following theorem and its analytic counterpart is proved. Theorem 2. Let \(F\) be a relatively compact set in \(\mathbb R_{+}^{d+1}\). The following properties are equivalent: (a) \(F\) is dominating in its closure for \(\text{BMO}(\mathbb R_{+}^{d+1})\). (b) \(F\) is dominating in its closure for \(\text{VMO}(\mathbb R_{+}^{d+1})\). (c) \(|\bar{F}\cap\mathbb R^d\setminus F_{nt}|=0\).