"There's a bear here"- is there a better way to say it?
"There" has a rather more complex set of meanings than simply "a place not near me". In this case it's being used as a placeholder for the delayed subject ("a bear") of the verb "to be" (being used in the sense of existing rather than as a copula). It has no sense of location to it in this usage.
So "There is a bear here." is equivalent to "A bear is (exists) here.", which you could use as an alternative if you prefer.
For more details: https://en.wiktionary.org/wiki/there#Pronoun