In approximately 50 years, NASA and many other international space agencies collaborate to create a cloud top base on Venus. The astro-geographers on the station (aka the loser's club) want to go to the surface to study the mineral formations. The people who know what they're talking about, however, say that they can't do that, due to the high temperatures. But the geographers say that they will dig into venus and find a dead lava tube to live in. My question is: Can they do this? (Not minding how to get there), and how deep would they need to go to have livable temperatures?
The two answers you already have are correct, but I think it is worth the time to add some more details.
After all, the idea is quite reasonable. Anyone who has been in a cave knows that it initially gets colder as you go down. Similarly, my native Florida is practically the land of springs, and anyone who has visited one knows that they are 72 degrees year round - quite refreshing during the hot summer days. We all know that it eventually gets hotter as you go deeper, but it initially gets colder, so why shouldn't there be a sweet spot on Venus?
The answer on Earth
Of course to answer that question we just need to know why it initially gets colder on earth as you travel deeper. The answer is that it doesn't really. Ground temperatures near the surface are actually just determined by the average yearly temperature of the region. The outer layers of the surface act like a big blanket, and of course there is a lot of ground underground (citation needed) so it changes temperature slowly. As a result, temperatures in the caves/aquifers that we are used to eventually reach and stay at the average surface temperature. In other words our Florida springs are 72 degrees because if you average out winter and summer Florida is 72 degrees. Therefore during the summer it is colder underground, but during the winter it is warmer.
The answer on Venus
As explained in the other answers though, the situation on Venus is much different. There is no winter and summer. Not even a day and night. While there is certainly weather on Venus and I'm sure the temperature varies as a result, it never reaches anything less than "near-instant death". As a result, nothing under the surface is ever colder than that either. There is nowhere reasonable to go.
Unless of course you really don't like your geographers. In that case an underground lava tube may be a great home for them.
Surface temperatures on Venus approach a toasty 900degF (~475degC). Though heat is radiated from the planet, there is no surface cooling as on Earth. Venus has volcanoes and presumably some amount of radioactive decay and probably a hot metal core, much like Earth.
There's no "sweet spot" anywhere close to that planet. Closer than your cloud top base.
As Arkenstein implied in their comment:
The surface temperature of Venus is uniform. It does not get any cooler as one approaches the poles, or at night, or any of the variety of things that change temperature here on earth. There is therefore no way for the temperature to get any cooler beneath its surface; any heat sink capacity has been used up for millions or billions of years since Venus' runaway greenhouse effect took effect.
So under the surface, it'd be hotter than the surface. The deeper you might dig, the hotter it would get.
There is no "sweet spot", no respite from the heat.
Despite what others said, temperature is not actually your problem. Pressure is. The pressure on surface is over 90 times what you want inside your habitats. Not to mention that pressure comes in the form of super critical carbon dioxide at 740 kelvins.
What this means is that the ratio of temperature difference is smaller than the ratio of pressure difference. So if you find a way to deal with the pressure, the heat won't be an issue.
I'd go for a plant that uses power to de-pressurize the supercritical carbon dioxide. Then give your habitat multiple pressurized shells with inner shells having progressively more survivable pressures. This is not mechanically as good as one shell but it would be easier to construct and if you fill the intermediate shells with de-pressurized carbon dioxide you will also get cooling "for free". For as long as the power plant works and carbon dioxide gets cycled anyway. You could also extract some water from the carbon dioxide. Probably not much but given that you'd want the habitat very well sealed anyway, maybe enough?
Some limitations on your power plant would apply though. If the power is produced with a heat engine of any kind or in general with a process that relies on temperature difference, you cannot use the cooling to create that temperature difference and then use the power produced to run the cooling. Well, you can, but you might have to name your colony the barony of Münchhausen for it to work.
Still, nuclear reactions can produce high enough temperatures for it to work even with cold end at 740 kelvins and other energy sources are possible. Maybe that supercritical carbon dioxide has flows that you can extract energy from?
In any case you'd probably need lots of power. And good redundancy with it.
The lava tubes are not really doing anything for you. Except trapping you underground and under a sea of supercritical carbon dioxide if things go wrong. I'd go for a rig or ship that floats in the carbon dioxide instead. Much easier to evacuate to those cloud top bases or to supply or build. And it can be moved.