到底〜ない
by no means, utterly (impossible)
到底 + potential negative / 〜ない
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到底 with a negative says that something is completely impossible, however you try: 一日でこの量は到底終わらない, "there's no way I can finish this much in a day".
It usually comes with a potential negative (できない, 信じられない, 払えない, 理解できない) or a verb of reaching, like 及ばない ("can't match") or かなわない ("no match for").
It's the formal equivalent of とても〜ない (N3). 全然 is casual and less about ability, so it sounds out of place in formal writing.
The kanji mean "reach the bottom": however deep you go, it won't happen. Don't use it with a positive verb.
Examples
一日でこの量は到底終わらない。
There's no way I can finish this much in a day.
彼の説明には到底納得できない。
I can't possibly accept his explanation.
そんな大金は到底払えない。
I could never pay that kind of money.
彼女の才能には到底かなわない。
I'm no match at all for her talent.
到底信じられない話だ。
It's an utterly unbelievable story.