I think some of the characters' ages are canon, although some are bit more wibbly wobbly. I really like that they are not all young and inexperienced
I think Spock was a lieutenant under Pike and was most likely mid twenties there, which would put him at least somewhere between Kirk and McCoy's age.
I'm not so sure about Chapel either. She'd had a scientific career prior to signing up, although admittedly we don't know for how long, and she was at least part way through an MD on top of that. She was most likely early thirties.
Rand was unofficially 24 in Miri (a line excised from the script) but I've seen some suggestion that a graphic in Flashback might have listed her DOB as being a year older than Kirk, similar to the real actors. Charlie X suggests that the latter is more plausible. She's also a senior non-com so a few miles on the clock makes more sense.
I've also seen a role-playing game that listed the ages as follows (presumably around the time of season 2)
Kirk 34
Spock 40
McCoy 40
Scotty 45
Sulu 29
Uhura 27
Chapel 31
Chekov 22
Rand 26 (or 35 if a year older than Kirk)
In the second season Kirk was 34 in "The Deadly Years", and the season was 11 years after he was a lieutenant according to "Obsession" and 13 years after he was a lieutenant according to "A private Little War". Kirk was a first year cadet at Sarfleet Academy 15 years before "Shore leave".
Chekov was 22 in "Who morns for Adonais?" but was probably aboard the Enterprise in time for "Space seed".
Sarek was 102.437 in "Journey to Babel" which meant that Spock could have been older than the rest - except that his mother Amanda looked a lot younger than 102! The script and The making of Star Trek say she was 58, about the age of Jane Wyatt. So Spock's age should be 58- X, X being Amanda's age when spock was born. Note that because Spock joined Starfleet he hasn't spoken to Sarek as son and father for 18 years or since Amanda was about 40. So Spock entered Starfleet Academy, or maybe graduated and was commissioned, when Amanda was 40. Spock served with Captain Pike for 11 years and was on the Enterprise with Pike, apparently the senior lieutentant after # 1 and third in command, 13 years before the first season episode "Menagerie".
Scott's age in "Relics" and the elapsed time give his age in "Generations" even though they seem to be alternate universes. That is not much help with his age in TOS.
McCoy was 137 in "Encounter at Farpoint" which can give his age in TOS if the interval can be correctly deduced.
McCoy and Scott were probably in their 40s in TOS, Kirk was 34 in "The Deadly Years" , Spock should have been older than Kirk ("Menagerie" & "Journey to Babel") but also should have been younger than Kirk ("Journey to Babel").
Chekov was 22 in "Who Mourns for Adonais?" Rand, Sulu, and Uhura should have been significantly older than Chekov and younger than Kirk.
It seems to me that the creators of the official chronology have not carefully considered the various factors which rather tightly constrain the ages and career dates of Kirk, Spock, and Chekov, and it is inaccurate in places.
Anyway, according to the official (but not canonical and somewhat flawed) chronology:
Scott was born in 2222.
McCoy was born in 2227.
Spock was born in 2230.
Kirk was born in 2233.
Sulu was born in 2237.
Uhura was born in 2239.
Chekov was born in 2245.
"The Cage" 2254.
"Where No Man has Gone Before" 2265.
First season TOS. 2266-2267.
Second season TOS 2267-68.
Third season TOS 2268-2269.
Thus in the second season the official chronology makes Scott 45/46, McCoy 40/41, Spock 37/38, Kirk 34/35, Sulu 30/31, Uhura 28/29, Chekov 22/23.