Stevenson's 2003 paper in Nature illustrates the scale of technology that would be required for such an undertaking.
Nature 423, 239-240 (15 May 2003)
Planetary science: Mission to Earth's core — a modest proposal
Abstract:
Planetary missions have enhanced our understanding of the Solar System and how planets work, but no comparable exploratory effort has been directed towards the Earth's interior, where equally fascinating scientific issues are waiting to be investigated. Here I propose a scheme for a mission to the Earth's core, in which a small communication probe would be conveyed in a huge volume of liquid-iron alloy migrating down to the core along a crack that is propagating under the action of gravity. The grapefruit-sized probe would transmit its findings back to the surface using high-frequency seismic waves sensed by a ground-coupled wave detector. The probe should take about a week to reach the core, and the minimum mass of molten iron required would be 108–1010 kg — or roughly between an hour and a week of Earth's total iron-foundry production.