I understand that the sentence means something like "This disease was eradicated 30 years ago", but I have no idea what role does 被 have in it?
Is the sentence okay if we remove it? Does it convey any special meaning when present?
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被 + verb = passive form
根除 = eradicate
被根除 = be eradicated
Some verbs have active form with passive meaning. 根除 is one of them. So it's fine to remove 被 from this sentence.
(These verbs are very similar to ergative verbs in English but mainstream Chinese grammar doesn't interpret them as ergative verbs.)
被 is the passive voice. It is the "was" in "the disease was eradicated.
But it is not the usual "was" in the sense of 是, which is a "state" verb.
Instead, the implication of 被 is, "was done."
It is OK to remove the "被”; we use both versions in daily life. If you add "被", you emphasize the passive form, and in this sentence, this kind of "passive form" can be omitted because the meaning of this sentence will not change if you omit the "被".
I think this is different from English. English is used to emphasize the relationship of object by grammar, but Chinese is used to emphasize relation by meaning. And in this sentence "病" was cured, it is "passive form", even you don't add "被", everyone knows "病是被动的", so you can omit it.
If you add "被", it is very very correct, but if you do not add "被"，no one can say that is wrong.