Place where something is done with

As a suffix, 先(さき) names the place or party connected to an action, as in 行(い)き先(さき) (destination) or 勤(つと)め先(さき) (workplace).

Further Reading

Examples

Place of lodging with 〜先

Manga panel from ご注文はうさぎですか? showing example of Place of lodging with 〜先.
ご注文はうさぎですか? » Volume 1 » Page 50

Since Cocoa helps out at Chino’s house where she’s saying, she considers the situation if she were instead staying with Chiya.

ココア:
「私(わたし)の下宿(げしゅく)先(さき)が千夜(ちや)ちゃんの家(いえ) だったらここでお手伝(てつだ)い させてもらったんだろうなー」
“If my lodging place were Chiya-chan's house, I probably would've been allowed to help out here.”

Place of part-time job with 〜先

Manga panel from ご注文はうさぎですか? showing example of Place of part-time job with 〜先.
ご注文はうさぎですか? » Volume 1 » Page 55

Syaro explain the qualities of a shop’s tea cup, but Rize points out that her workplace primarily sells coffee.

シャロ:
「リゼ先輩(せんぱい)のバイト先(さき)行(い)ってみたかったのに…」
“I wanted to try going to Rize-senpai's part-time workplace...”
ココア:
「もしかしてコーヒー苦手(にがて)?」
“Do you not like coffee?”

Staying with relatives with 〜先

Manga panel from ふらいんぐうぃっち showing example of Staying with relatives with 〜先.
ふらいんぐうぃっち » Volume 3 » Page 34

Makoto introduces Chinatsu and Kei to Anzu while the group is meeting one another.

真琴(まこと):
「こっちの二人(ふたり)は居候(いそうろう)先(さき)の親戚(しんせき)で千夏(ちなつ)ちゃんと圭(けい)くんです」
“These two are the relatives I'm staying with, Chinatsu-chan and Kei-kun.”
千夏(ちなつ):
「こんにちはっ!」
“Hello!”
圭くん(けい):
「どうも」
“Hi.”
杏子(あんず):
「椎名(しいな)杏子(あんず)です よろしく」
“I'm Shiina Anzu. Nice to meet you.”