Infinite direct product of fields.
Solution 1:
The dimension is the cardinality $|F|^{\aleph_0}$ according to the Theorem of Erdös-Kaplansky. See MO/49551 for a general formula of the dimension of an infinite family of vector spaces. All this belongs to "set-theoretic linear algebra", and no bases can be explicitly written down, since the axiom of choice plays a central role. Probably there are models of ZF where $\mathbb{Q} \times \mathbb{Q} \times \dotsc$ is no direct sum of copies of $\mathbb{Q}$; hopefully a set-theorist can add a reference for this.