Can replication and translation emerge in a single mechanism via self-assembly? The key molecule, transfer RNA (tRNA), is one of the most ancient molecules and contains the genetic code. Our experiments show how a pool of oligonucleotides, adapted with minor mutations from tRNA, spontaneously formed molecular assemblies and replicated information autonomously using only reversible hybridization under thermal oscillations. The pool of cross-complementary hairpins self-selected by agglomeration and sedimentation. The metastable DNA hairpins bound to a template and then interconnected by hybridization. Thermal oscillations separated replicates from their templates and drove an exponential, cross-catalytic replication. The molecular assembly could encode and replicate binary sequences with a replication fidelity corresponding to 85-90 % per nucleotide. The replication by a self-assembly of tRNA-like sequences suggests that early forms of tRNA could have been involved in molecular replication. This would link the evolution of translation to a mechanism of molecular replication.