In Modern Japanese, one can use "がある” and "がいる” to express "There is/are ...". The existential sentence can be further used to assert the presence or existence of something in a particular place: "[Place]に[Something]がある/がいる".
How to write existential/locational sentence in Classical Japanese?
What you're calling existential and locational sentences could be written in a number of ways in classical Japanese, as in modern; for example, in both CJ and MJ there are honorific and humble verbs that can be used. If you just want a basic, simple equivalent for ある and いる, the classical Japanese verb あり will do for both. It was used with both animate and inanimate subjects, as follows (notice the absence of が):
Existential: [someone/something] あり
Locational: [place] に [someone/something] あり
In practice, of course, あり was often combined with one or more 助動詞.
Some examples:
Existential (animate), from 『竹取{たけとり}物語{ものがたり}』
今{いま}は昔{むかし}、竹取{たけとり}の翁{おきな}といふ者{もの}ありけり。
Locational (inanimate), from 『方丈記{はうぢゃうき}』
また、麓{ふもと}に一つの柴{しば}の庵{いほり}あり。
Locational (animate) , from 『方丈記{はうぢゃうき}』
かしこに小童{こわらは}あり。
Over time, the verb 居り{をり}, which originally had a different range of meanings (including things like "to be still" and "to sit"), also came to be used as an existential and locational verb with animate subjects – as did 居る{ゐる}, which started with an even broader range of meanings.
