"need not to be neither atomic nor ordered"

Solution 1:

No. The sentence as written is over-negated. You can't use "not" and "neither" together; you want "not... either."

"Need not to be" is also grammatically incorrect.

We can say "don't need to be." The uncontracted equivalent of that would be "do not need to be." You need to use the auxiliary "do" because it is negative.

If you don't include "do," you can use a separate construction, "need not be," where "need" functions a bit like an auxiliary. In this construction, you do not include "to" before the infinitive.

Solution 2:

As with most Boolean expressions, you can factor this out to be:

Ordered operations require that space on disk is allocated sequentially, so that spatial ordering corresponds to the chronology of operations, but actual disk writes need not be either atomic or ordered.