Think of something standing at the lowest line it can stand on and not falling. That is the picture worth holding for subsist. Put a person there, with one kind of food and almost no money, and you get the everyday meaning: they are alive, only just, and nothing more than that. Take the person away and leave only the standing, and you get the legal meaning: a right or a duty is still up, still in force, even though time has passed and other things around it have ended.
From the word’s history
The packet's Wiktionary entry derives subsist from Latin subsisto, a halt or a standing, built from sub (below) plus sisto (to stand or place). The word therefore begins as standing still down at the bottom, which is close to both modern uses.
packet 裡的 Wiktionary 把 subsist 追到拉丁文 subsistō（停住、站住），由 sub（在下面）加 sistō（站、放）組成。所以這個字一開始就是「在底下停住、站著」，跟今天兩個用法都對得上。
Wiktionary entry supplied in packet (source_ref kaikki:3f6f69a190036ae4:1097719657)
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