。
I'm not healthy enough to be taking care of other people.
Literal
I [topic-は] person [attributive-の] care [object-を] can-burn-[to-the-extent-ほど] healthy is-not.
~ほど ('to the extent of X') is used here in the pattern [verb potential]+ほど+[adjective negative] to mean 'not X enough to do Y.' 世話を焼ける is the potential form of 世話を焼く ('take care of'), an idiomatic verb+object expression where 焼く is fossilized — unrelated to its literal 'burn' meaning. 元気 ('healthy, energetic') is a common na-adjective. The speaker is declining to look after others because they themselves aren't well.