Octopuses have 8 tentacles, spiders have 8 legs.
Is there something special about 8? It seems like an animal that needs 360° mobility has 8 legs.
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This is just a wild guess, but for movement on a surface in which legs are lifted from the surface, six legs might be sufficient to provide three points of support at all times (think three-legged stool) while allowing relatively rapid movement (legs on both sides can be moved simultaneously), but such might be both less stable and less friendly to variable direction of movement (since legs are in fixed locations and not evenly/radially distributed in animals with bilateral symmetry, having more legs would facilitate balance for different directions of movement). Having eight legs allows well distributed support and might allow more simplified motor control.
Having more than eight legs would tend to lengthen the body (for animals with bilateral symmetry) which might tend to bias movement direction (as well as possibly being wasteful of resources).
There might also be correlation of highly variable movement direction with requirements for greater stabilization. E.g., eight legs might provide greater stability on a web (which surface might encourage more variable movement directions) and better support large claws (which might make sideways movement more useful in face-to-face encounters).
Note that the above does not address non-walking animals nor animals with radial symmetry (For an animal with radial symmetry, six legs might be sufficient to provide support with three.).
Hopefully, a better answer will be given, since I have not even paid attention to how spiders (or crabs) move. (The manner of movement would give insight into how significant support/stability and motor control are.)
First of all, it's wrong to assume that an organism "needed 360 mobility" and therefore ended up with 8 legs, as the question implied. That would be a misunderstanding of evolution.
No, there is nothing special about 8 legs. Might as well be 6, 10 or 12. Take note, it would have to be an even number all organisms you mentioned are bilaterally symmetrical.
However, for these organisms 8 limbs were obviously something special. But independently, as they aren't closely related - octopusses are molluscs and spiders are arthropods. Maybe octopusses have 8 limbs because it gives them the best moving speed and agility while not expending too much energy. It's always a delicate balance.
The difference between an insect (6 legs; 3 segments), a spider (8 legs; 4 segments) and a lobster (10 legs; 7 segments) - all of them athropods - is in the number of leg segments. By the way, molluscs aren't considered segmented so they don't fall into this category.
When it comes to the difference between 8 and 6, You don't know did spiders "get" an extra leg segment or did insects "lose" one in order for the legs to be less of a burden during flight.
Interestingly, evidence suggests insects are the last to appear in the evolutionary family tree. Spiders also seem to have ended up with 8 legs by reduction as opposed to addition.
To answer the question "why do some animals have 8 limbs" - they have an even number of limbs because they're bilaterally symmetrical. The exact number of limbs in molluscs is a balance of energy consumption and fitness, as always, and the number of limbs in arhtropoda is the same thing, and my personal opinion is that spiders didn't go beyond reducing the number of their limbs to 8 for some reason - maybe because with 6 they were less efficient in walking the spiderweb.