Guide: Hiding your base
This guide is most useful for those who have read the Newbie Guide, survived spawning and have established a basic base. It will try to teach you how to best increase your chances of surviving by making your base difficult to find.
You likely already have a basic base but it is probably small, built in a rush and closer to spawn than is generally considered "safe". Step 1 of surviving well is to have a difficult to find, hidden and well equipped base.
Coordinates for your base
With the current map being 5k blocks from spawn in each direction there is a lot of room to decide where to set up base, the minimum “safe” distance is usually considered around 3k from the spawn point on any axis. However, setting up right on the edge of the world also isn't a great idea as often base hunters search there more thoroughly than other areas simply because it's often considered the “safest place”. A base somewhere between 3-5k on one axis (preferably both) is probably the safest bet.
Each map we often find one quadrant more popular than the others. Make note of this trend as you explore and decide if you want to be closer to others or more distant.
The entrance to your base should be well hidden, the best hidden entrances are usually under an ocean or large lake, covered by at least a two block thick layer of the surrounding material. If you decide to make your base entrance on land under some grass blocks remember that it may take some time for grass to grow over newly replaced dirt blocks. The next person to wander through your territory may see the dirt blocks and destroy them, revealing your entrance. It's therefore best to wait at least 10 minutes before leaving the entrance area or logging off to make sure the grass has regrown. Another option is to lure some sheep over the entrance with wheat to try to disguise it, although this may also draw players looking for wool for builds or experience from passive mob kills. Alternatively, you can avoid this problem by never removing blocks you cant replace (ie, make your base in the side of a dirt (not grass) wall.)
No weak points
The base shouldn't be connected to any caves or abandoned mineshafts. These can be found and explored and if one leads to your base, or even just close to your base you could be in danger of being found. The presence of extra torches placed during your exploration can give away your presence. Seal off caves with smoothstone and make them look natural to avoid suspicion. Some people also say spawn is better, but if you think about it, no one builds close to spawn (as in a good 1000 blocks away) so in a way making a base there is a decent option as well.
To be really safe your base should be built with its lowest point at Y layer 18. Diamonds occur up to Y layer 16, mines that check layer 16 will therefore miss mining into your base - they will go right under it.
Things such as passive mobs being farmed, automatic piston looping machines and flowing water make noise. Miners and cave explorers may decide to investigate this noise and dig towards it potentially finding your base. If you are going to include these things then take into account the distance at which they can be heard. Tamed wolves are notorious for giving away hidden bases.