Btukfyl

In Star Trek IV: The Voyage Home, the Bounty travels back to what looks to be Earth circa 1985 to acquire a breeding pair of whales. Yet at this time, whales were already in danger and people were sophisticated enough that the Bounty might be detected and possibly destroyed.

Why did they not travel back to 900 AD or so before mass whaling or missiles became a thing?

This is addressed in the film's official novelisation which contains a slightly extended version of Kirk's conversations on board the Bounty. Mr Scott was making the plans to turn the hold into a holding tank, per the Captain's orders and evidently worked out that he'd need materials from twentieth century (onwards).

"Scott here, admiral. Wi' the proper materials--the proper twentieth-century materials--I'll be able to build ye a tank."

"Thank you, Mr. Scott."

Coupled with the known extinction date of the humpback whale, this gives them a pretty solid date range to aim for with their time warp maneuver. Although they had quite a broad range in mind ("late twentieth century"), Spock's calculations turned out to be pretty accurate. Presumably he'd have been just as happy with 1975 as 1999.

"Ready to engage computer, admiral," Spock said.

"What's our target in time?" Jim asked.

"The late twentieth century."

"Surely you can be more specific."

To transport the whales the crew used transparent aluminium for their makeshift holding tank which they had to "invent" themselves and produce using 1985 technology which wouldn't have existed in 900 AD.

I'm not completely familiar with the "time sling-shot" maneuver, but there are multiple possible reasons for that.

Going further back is dangerous

It might be that ~1980 is roughly the earliest year they can safely travel. Traveling any further back might strain the Bird of Prey (also dubbed "rust bucket") too much and you all might die. Any further into the future and you might not have the chance to encounter whales.

Time travel is not an exact science

From what we can see in the movie they never specify to what year they want to go. Quite the opposite, directly after the jump this dialog happens:

Kirk: Picture, please.

[Picture of Earth appears in the viewscreen.]

Kirk: Earth. But when? Spock?

Spock: Judging by the pollution content of the atmosphere, I believe we have arrived at the latter half of the twentieth century.

So they know how to roughly jump back in time, but they can't target a certain year or period.

Time travel is an exact science

In the TOS episode "Tomorrow is yesterday" the crew of the Enterprise learns the mechanics required to jump through time by accident. They complete a maneuver around a black star which hurls them back into the ~1960s on Earth. At the end, they can execute the same maneuver around Sol to travel back to their time.

Unbeknownst to them, this time travel method might have a fixed time span of roughly ~300 years you can jump backwards and forwards with, as the dates neatly line up. Or maybe they knew about it, but it was never established on screen.

Repairs and Extensions

Last but not least, the Bird of Prey is ill equipped to transport whales. That means that you need the resources to remodel it on the fly in the past. That excludes anything before ~1960 I guess, because since then we have made vast advancements in technology. Additionally, Earth is in a pretty bad shape starting from ~1991 with Eugenic War waging and killing tens of million of people, and undoubtedly binding a lot of resources. So you want to land before that, because afterwards you'd either end up in one of three situations:

• You can't acquire the resources required because they are not available because of the war.


• Whales are already extinct.


• The technology has advanced so far that you have a hard time doing your thing.

With these restrictions in mind, the period around ~1980 sounds like quite a good place to be.

They had a narrow timeframe that would be useful to them.

• They needed to arrive late enough that the whale-tank materials they needed to modify the ship would be readily available -- post-1950s. Not just "possible", but in stock locally.


• They needed an economy robust and entreprenurial enough that their oddball material purchases needs wouldn't pop up any red flags on any government's radar (for instance doing this in the Soviet Union was right out) - so 1950s and up.


• If they had any problems with the vessel's drive systems (and they did), they needed a modern enough society where technology, electronics, nuclear things, etc. were well-developed and readily available (so 1970s at least, and the later the better!)


• 
  Whale science needed to be solidly established, with plenty of experts available - because they needed to catch whales, and they had little chance of getting Klingon sensors to tell them the difference between a gray whale and a sperm whale.
  
  
  
  
  • Even better if culture had become affluent enough that there were aquatic zoos like Sea World, so we're definitely into the 1970s.
  
  
  
  


• Preferably, racism (including war sentiments) have declined considerably, so Uhura, Sulu, Chekov and Spock will have an easier time, so 1970s and the later the better. It would also help if the Cold War was abating, so 1980s.


• Can't go later than 1991, because in Trek canon, the Eugenics Wars began.