Lategame?
Yes, an ADC will kill a squishy target in 2-3 auto-attacks. That damage output is the very thing they are designed to do.
But you can get a defensive item, even on an assassin. For instance, new Thornmail works rather well:
Level 18 Assassin has around 85 base Armor. Thornmail + runes are 84 bonus Armor. Total, 169 Armor, or 140 against a Last Whisper Upgrade.
For health, let's take 2k base, 250 from Thornmail, 250 from Edge of Night. Total, 2500 HP.
EHP: 6725 / 6000 against physical damage.
Baseline ADC:
100 base AD, 0.650 base AS, 3% AS/lvl (51% @18), 2k HP. 15 AD and 17.5% AS from runes and masteries.
Build 1:
390 AD, 1.58 AS: 750.75 Damage-per-attack, 1186.19 DPS
Build 2:
325 AD, 1.52 AS: 786.5 DPA, 1195.48 DPS. On average, 352 magic damage every 4 attacks (guesstimate).
Build 3:
380 AD, 1.39 AS: 665 DPA, 924.35 DPS
Build 4:
340 AD, 1.75 AS: 748 DPA, 1309 DPS. On average, 352 magic damage every 4 attacks (guesstimate).
Looking at these builds (and ignoring Shiv damage for now), the Thornmail-possessing assassin would survive, on average:
Build 1: 7.99 attacks, 5.21 seconds.
Build 2: 7.63 attacks, 5.42 seconds.
Build 3: 10.11 attacks, 8.46 seconds.
Build 4: 8.99 attacks, 5.38 seconds.
Well, how about an utterly squishy assassin, then? Assuming he didn't build any Armor?
85 Armor base, 94 total (runes), 91 against LWU. 2250 HP (still has EoN).
EHP: 4365 / 4297.5
Build 1: 5.99 attacks, 3.16 seconds.
Build 2: 5.72 attacks, 3.29 seconds.
Build 3: 6.56 attacks, 4.32 seconds.
Build 4: 5.84 attacks, 2.86 seconds.
TL;DR: Build like a squishy, get popped like a squishy. Building a bit of Armor goes a long way to help your survival.