No, it is not feasible to dig a hole to the center of the Earth.
The deepest such hole appears to be the Kola Superdeep Borehole, which goes down about 12km, or about 0.2% of the way to the center of the Earth.
The pressure at the center of the Earth is so high that rocks are about five or six times denser there than at the crust; they have literally been crushed down to a fraction of their original size. The temperature there is believed to be around 7000K, hot enough to melt just about any metal. Clearly, this is not a reasonable engineering goal.
You might be interested in a 2003 paper by Dave Stevenson proposing to study the deep interior of Earth by dropping molten iron down a crack in the crust and allowing it to sink all the way to the core. I can't seem to find the original paper, but here's a readable summary. The reference is
- A87. Stevenson, David J. Mission to Earth’s Core -A Modest Proposal. Nature, 423, 239-240, 2003.