Old science fiction had lots of stories that involved people exploring the center of Earth, like for example Journey to the Center of the Earth, by Jules Verne. As Earth sciences advanced, this sort of story fell out of favor, and today even sounds somewhat ludicrous, as now is well understood pressure and temperature increase fast as one descends into Earth, so it's unfeasible to dig and hold open large cavities anywhere near the core, much less survive for any period of time there. So, despite the fact Earth radius is close to 6400km, the deepest mine in the planet, Mponeng Gold Mine, goes only 4km under ground level, or 0.0625% of the distance required to reach the center of Earth. I don't know how deeper could it go, and if the limiting factor precluding greater depths is heat removal due to high temperatures or tunnel collapse due to overburden pressure, but I don't believe it can go much beyond current depth.
But if our planet is too large for this scenario, making it infeasible here, surely there is a lower threshold for mass and size making it actually possible. So I'd like to know, what's the maximum size for a planet/dwarf planet/asteroid, where would be possible in principle to dig tunnels all the way down to the core, or perhaps even to build some sort of "corescrapper", an underground multi-floor building going from surface to the core? By tunnels I mean tunnels that people could actually pass through, using technology similar to what is used in current mines here on Earth. So just being drillable, like in a oil well, to the core, doesn't count.
If necessary, the putative celestial body can be assumed to have rocky composition, perhaps with a metallic core, and age similar to the current age of the solar system.