We prove that there is a normed barrelled space with dimension $\mathrm{non}(\mathcal M)$, which denotes the smallest cardinality of a non-meager subset of $\mathbb R$. Consequently, it is consistent with $\mathsf{ZFC}$ that there is a normed barrelled space with dimension $<\!\mathfrak{c}$. This answers a question of Sánchez Ruiz and Saxon. This is a special case of a more general theorem: for every infinite cardinal $κ$, every Banach space with density character $κ$ contains a barrelled subspace with dimension $\mathrm{cf}[κ]^ω\cdot \mathrm{non}(\mathcal M)$. We also prove that if the dual of a Banach space does not contain $c_0$ or $\ell^p$ for any $p \geq 1$, then that spaces does not have a barrelled subspaces with dimension $<\!\mathrm{cov}(\mathcal N)$, which denotes the smallest cardinality of a collection of Lebesgue null sets covering $\mathbb R$. In particular, it is consistent with $\mathsf{ZFC}$ that no classical Banach spaces contain barrelled subspaces with dimension $\mathfrak{b}$. This partly answers another question of Sánchez Ruiz and Saxon.