Does anyone know how old Emperor Palpatine was? He seems about 60 in Episode 1 and about 70 in 2 and 3 (I'd guess) so pretty old in the original trilogy - 90 ish? And, I think this counts as the same question as presented in my title, does the Dark Side or the Force in general sustain him? It just occurred to me that some anti-Empire groups might just comfort themselves thinking, "He won't last much longer."
To the best of my knowledge Palpatine's birth year has never been mentioned in-universe anywhere. However, an archived blog entry on starwars.com written by Leland Chee lists the major characters' birth years. It states that Palpatine was born in 82 BBY, and that Return of the Jedi took place in 4 ABY, so Palpatine was about 86 years old when he died:
~ -600 Jabba the Hutt
~ -200 Chewbacca
-102 Count Dooku
-92 Qui-Gon Jinn
-82 Palpatine [47BrS:8:11]
-72 Mace Windu
-66 Jango Fett
-57 Obi-Wan Kenobi
-46 Padmé Amidala
-41.9 Anakin Skywalker
-32 Star Wars: Episode I The Phantom Menace [3:4:14]
-31.5 Boba Fett
-31 Lando Calrissian
-29 Han Solo
-22 Star Wars: Episode II Attack of the Clones [13:5:21]
-21 Wedge Antilles
-19 Star Wars: Episode III Revenge of the Sith [16:5:23]
-19 Luke Skywalker & Princess Leia (Luke born first)
-18 Corran Horn
-18 Mara Jade Skywalker
0 Star Wars: Episode IV A New Hope
3 Star Wars: Episode V The Empire Strikes Back
4 Star Wars: Episode VI Return of the Jedi
9 Jaina & Jacen Solo (Jaina born first)
10 Anakin Solo
26.5 Ben Skywalker
Palpatine indicated in Revenge of the Sith that his master, Darth Plagueis, could create life and prevent death. However, Palpatine apparently did not learn how to do it before he murdered Plagueis; when Anakin became Darth Vader, Palpatine told him that he did not know how to prevent death (which would presumably involve the ability to prevent aging):
Anakin: Just help me save Padme's life. I can't live without her. I won't let her die. I want the power to stop death.
Palpatine: To cheat death is a power only one has achieved, but if we work together, I know we can discover the secret.
So did Palpatine discover how to cheat death using the dark side between Revenge of the Sith and Return of the Jedi? From the linked question the conclusion is that, canonically, there is no indication that he did; however, there are some Legends sources in which he did figure it out.
How old was Palpatine?
Palpatine was born 82 BBY and died 4 ABY
So he was ~86 when he died
Did the Force help sustain the Emperor in his old age?
There is no direct evidence claiming the Force was used to help him live long (and prosper) but there is some evidence that the Force can be used to sustain life:
The dark side puts his body under a lot of stress. He used clones to keep himself fresh but those cloned bodies went to pieces faster than his original body. In Confrontation on the Smuggler's Moon Palpatine tell Leia, "Long ago I found my flesh could not withstand the awesome demands of the dark side... the great Emperor Palpatine discovered he was dying. My body was literally consumed by the energies I had released... fortunately... fortunately there was a way... a road to eternal life." It seemed his master made some ground in keeping his body from aging. See the novel Plagueis.
The scars that had grown over his wounds had abruptly begun to soften and fade, and he had begun to breathe more freely than he had in twenty years. He began to sense that not only were his damaged tissues healing, but his entire body was rejuvenating itself. Beneath the transpirator, areas of his skin were smooth and youthful, and he knew that eventually he would cease to age altogether.
According to Wookieepedia and similar source-material, he was 86 at the time of Episode VI - Return of the Jedi. His "used and dried" appearance originated from the dark side (it takes its tribute) and of course from the fight with Windu (Episode III).
In beyond all canonic sources I have heard speculations that he is about 130-150 years of age, or even older - one even going so far to suggest he is over 800 years old. But thats all speculation, I stick with what the creators say.