Assumptions
Facility run by computers, facility constantly repaired and upgraded by computers and robots. Lots of tiny repairing spider robots.
Lots of computers = lots of heat.
Lots of brains in fluid in glass cases, can’t have the precious cases breaking. How long does a brain last out of its case?
Brains need oxygen rich water and some sort of nutrient mix/bath to survive.
all sites can be targets for sabotage (not necessarily will be, if everyone is a brain, in a virtual reality...who is doing the physical sabotage?)
You have four main types of regions to look at for your site/s location…
(1) Space
Already mentioned…Have all the benefits of vacuum and zero-gravity
BUT
Space just doesn’t add up
(2) Air
You can have large floating air depositories for your brains. This will help avoid your problems with volcanoes, earthquakes shaking things up on the ground in your brain facility. These could be designed in any shape that is feasible/ imaginable but things to take into account:
is your platform active or passive? ie does it float where the wind blows it or does it have control of it's actions?
does it have big motors or does it have sails and big balloons of helium/hydrogen? Note: helium has less lift but is non-flammable. While hydrogen has more lift, but is kinda flammable (understatement of note!). Helium is also less abundant on Earth, as it escapes into space, which is why the airships never really took off in the past (pun intended).
does the motors and computer equipment require fuel? what sort of fuel? solar would be better than relying on oil or natural gases. Especially if humans stored as brains is the result of overpopulation and resource scarcity.
Altitude adjustment to avoid turbulent and stormy skies AND volcanic ASH clouds. Ash plays havoc with airplane motors which I learnt watching Dante's Peak as a kid, and the rest of the world learnt when Eyjafjallajökull erupted in 2010.
Ensure that all sensitive equipment (and brains) are not fried if the platform/s are hit by lightning. Also helps if your ship is not too fill of flammable gases.
If your air depositories have active control, they can guide themselves to landbased depots to pick up new brains. If your air depositories are passive and have no control of their actions, you will need drones to send/receive brains from all over the globe.
So, all this kindof rules out air as a safe spot.
Extra thought, you don't have to have your brains floating very high above ground to avoid the effects of earthquakes. You could have them all attached to small balloons (or more practically a suspension system) so that they all hover a few centimetres to metres off the ground. I think you will still need to make sure the roof doesn't collapse on them, if there is a powerful earthquake.
(3) Land
As mentioned by you and other answers,
try and avoid earthquake prone areas (California is out)
try avoid volcanic prone areas (Japan and Hawaii - Pacific 'Ring of Fire' is out). This includes anywhere in the vicinity of yellowstone national park.
try avoid rugged mountainous terrain as transporting tens of thousands of new brains a day will be treacherous!
try avoid river plains as prone to excessive flooding every 100-1000 years
try avoid coastal areas prone to changes in sea level!
be aware that some land areas are actually rising/sinking/moving horizontally due to Glacial Isostatic Adjustment. For example, Scotland is rising, but southern England is actually sinking in response. The interior area by the North Sea is experiencing lift/rebound, so the coastline is actually sinking (sea level rise is faster than the rest of the world in these areas. Regardless of human induced changes or natural changes due to, among other things, the post glacial rebound).
- If your facility has to last for tens of thousands of years, try avoid areas that may experience Glaciers! They carved up mountains, I doubt a few man-made structures will stop them.
Your machinery and computers will be creating lots of heat, so try avoid already hot locations like Australia or most parts of Africa, unless you have sufficient excess energy for lots and lots of air-conditioning.
Most Machinery rusts if exposed to oxygen and water, so try avoid moist tropical locations with lots of humidity/rainfall
Deserts while dry, will need sufficient excess energy to cool down your machinery. Stored water would be best for this situation!
Does the machinery and computer equipment require fuel? what sort of fuel? solar would be better than relying on oil or natural gases. Especially if humans stored as brains is the result of overpopulation and resource scarcity
- In summary, don't go to too high a latitude as glaciers are a risk in the next few thousand years, don't go to the equator and tropics because it's too humid, don't go along the coast lines because of sea-level changes, don't go into the interior because lack of water to cool machines/computer down, don't go to the mid-latitudes as most of the large deserts are located there.
Essentially go to ...
(4) Water
I had already decided water would be the best place for your facilities when I saw the answer about the 1,5 million manmade whales! The point about the krill and plankton as a nutrient source was inspirational!
Water covers 73% of the Earth's surface, and that is not taking into account the volume of space the oceans are taking up! Create depositories that can float on/under water. This will negate the need to worry about earthquakes.
Tsunamis, normally created by earthquakes, are only an issue when the shockwave travelling through the water reaches shallow coastlines. You do have reports of those freak standing waves that will overturn your Poseidon, but if you have your depository in a modular format (like a set of railcars) you should be able to survive the transit of the wave underneath you!
Volcanoes already on land aren't too much of a problem in the sea, just ensure your floating depository doesn't barge into any areas where the lava is entering the sea. E.g. the Hawaiian Islands active volcanoes. Sea mounts erupting from the ocean depths, should be noticed by ships equipment beforehand; you shouldn't have any nasty surprise eruptions right underneath you! You should have time to row away...
Sea level is 'fairly equal' all over the world and any change in sea level won't affect your facilities out at sea. It's only a problem to those on the coastline!
The surrounding sea water can be used for cooling systems and provide water (fresh if you include a desalination plant on board). Salt DOES collect and jam up machinery, but if the robots looking after the brains can keep on top of it, they should be able to keep all vital equipment running ship shape (pun intended).
When bad weather comes in, and it will. You can have your depositories lower themselves to a depth below the wave height and just sit out the storm in peace and tranquillity.
Your machinery can use a combination of solar, and wind energy. They can also utilise the energy from the ocean waves themselves. Hey! If you have brains in jars, and interactive virtual reality...you can figure out wave energy!
TOTAL SUMMARY: There are pros and cons for wherever you choose to place your facilities; this is probably why they are safest in our bodies! I figure centralised transit depots will be required for whichever option you end up taking.
I am for the sea, then hovering, then land, then air, and then space.