The ancient Greeks didn't do any trans-oceanic travel anyway, so the question is arguably moot.
Before the armada days... there essentially was no ocean travel. Yes, brave Norse settlers made it to the Americas, but the travel was lengthy, arduous and risky enough that only a few made it to Greenland and a little beyond, and even they arguably got lucky: if North America had been a little more southerly, all they'd have found would've been sea ice.
[Greek trade routes, via Ancient History Encyclopaedia]
So really, you don't even need to make any changes, apart from wiping out those pesky Norse adventurers, and making it so that the continents didn't have any convenient paths between them, like "hey if you just follow the edge of the sea ice west from Norway, you get to a McDonald's!".
However... that's not very "hard science" of me. In fact, science (Feasible Ocean Routes to and from the Americas in Pre-Columbian Times, quoting Chronica-Botanica-Vol-xiv) says:
It would be foolish to assert that there were no communications across the Pacific in pre-Magellan times
I feel that there is a fairly solid case for this being at least a thing that was feasible to happen at that time, given that hundreds of plant species have been found which appear to have spread between the continents with no other explanation.
We want ten thousand years of no seafaring, so we need defense in depth! Many layers! Travel between continents only when sea levels change and expose land bridges!
So while it might be a helpful factor, merely saying "OK, there are no easy paths, and no brave adventurers" won't cut it :)
Making seawater scary with predators, as rek suggests... again, it might be a useful additional tactic, but I don't think it would work, not for thousands of years for humans, anyway.
Waterfront property has been prime property as long as humans have existed. We migrated along waterways.
We take "dangerous" as a challenge. We've obliterated almost all large animals that we don't deliberately breed to eat, and all existing species are getting smaller as we kill and eat the biggest ones. For millennia we have harvested food from waters filled with shark and piranha and crocs and gators and vipers and poisoned sea urchins and deadly jellyfish and... and we eat them.
This isn't a new habit: we probably killed off the NA mammoth and smilodon, and likely were a contributing force for much of the rest of the wave of Late Pleistocene Megafaunal Extinctions.
So, sure, have predators, but just as one of many reasons the sentient land-dwellers ("They") have, to avoid screwing with the ocean.
The most valuable kind of predator could be the ship-attacking kind. If any large craft going past the continental shelf runs a very good chance of getting capsized boned by aggressive or amorous whales, then most craft won't go past that line, at least until they realize how much ambergris, whalebone and whalemeat are worth.
Hrm, OK, we don't want to give them an economic incentive to venture out: maybe kraken, rather than whales, then.
Nutrition vs Poison
(I see @Kari beat me to this one, darn! Updoot for Kari!)
On Earth, we may well have taken to the water due to an aquatic phase in our evolution... but whether or not that is the case, we stuck by the water because it's a fantastic source of nutrition, such as DHA, an omega-3 found abundantly in human brains and in aquatic foods. This aquatic aspect of our diet may well have contributed to our large brains. Oh, and salt, we use a LOT of that, with our abnormal behaviors like sweating and crying.
If everything in the ocean was toxic to Them, They'd have no real interest. Add some heavy metals, radioactives, some lead and arsenic and some faster-acting things, to the seaweeds and shellfish, and have the toxins concentrate more and more up the entire food chain.
Perhaps make rock salt toxic to Them. That's extremely believable, given how toxic it is anyway. Or have other soluble minerals leech out of the rocks of the land and into the oceans.
But you don't even need to specify what toxins affect Them; just that sea animals have the ability to process ocean foods that They as land animals have since lost. Remove the omega-3s, or remove Their need or ability to process them.
With no seals, polar bears, edible fish, etc to catch and eat, and nowhere to stop and find freshwater or meltable snow, travelers would be constrained by supplies they could take with them. With a large enough distance between, this would be a hindrance, and set a hard date for having to turn around, equal to less than half the provisions a craft could carry.
We can swim and float, so can spend time in water, make dugout canoes that can tip us in without dying, and so on. Let's get rid of all that. Their species sinks in water. Heavy bones, dense muscles, small lungs, non-closing airways, no diving reflex, poorly formed for swimming... so not real fond of swimming in the first place, even less fond of swimming anywhere near the deep, toxic ocean.
As L. Dutch pointed out, if the only buoyant stuff They ever see is bloated corpses, They'd take a while to even come up with the idea of floating airbags for travel, and those just aren't very ocean-worthy, but do well enough on rivers if you stick them under some kinda raft framework. Waterway dirigibles, pulled by draft animals trudging along the waterside, as we did with canals; or by pulling on a rope alongside the waterway.
It's admittedly a "small leap" from there to surrounding them in a solid sealed box instead of a mere framework, and from there to the discovery that it doesn't matter if the solid box has airbag in, or even whether it has holes in, so long as they're above the waterline, and from there to realizing that you can just float an open-topped box in a way that it won't flip over and drown everyone, and from there to ballast and sails and masts and rudders and rigging and then reinventing rigging to sail into the wind...
But even so... look how long it took us to develop wheeled vehicles (5500ya)! And another three thousand years after that (2400ya) to put teeth on it and make a cog, a thousand years after that (1300ya) to make a clock, and six hundred years more (700ya) to make a worm gear! Each of these, looking back, is a "small leap". And none of them kills us much if we screw it up. And even then, the wheel didn't really penetrate into sub-Saharan Africa until the Europeans turned up in the 1800s.
Hot air on land goes up. This means wind blows from the sea: a "sea breeze". In the night, it blows the other way, as the land cools, but the sea stays warm. So Their more sensible sailors make sure to be home before that happens, rather than get stranded away from shore.
Why would they be stranded? Because with poor rigging tech, They couldn't tack into the wind very well.
If you also have a climate without any constant trade winds for their navigators to follow, the inconsistent and mercurial climate would mean that there'd be no obvious paths for people to try exploring in. I'm not sure how realistic a world without tradewinds and currents would be, though...
The Armadas of Britain, Spain, France, and Holland, were all formed from forestry. The devastating impact of the formation of the British armada on the countryside of the UK can hardly be overstated. Maintaining the armada required the formation of a proper forestry commission with control over sizeable tracts of land doing nothing but the slow process of growing trees long enough to lay straight keels, masts, and so on.
I have heard that in the great expedition of 1588 it was expressly
enjoined the Spanish commanders of that signal Armada that if, when
landed, they should not subdue our nation, and make good their
conquest, they should yet be sure not to leave a tree standing in the
Forest of Dean.
If your civilization lacks wood as a construction material, and instead uses mostly stone, leather and metal, then that both sets a cool aesthetic, and helps to explain why large ships might not get built for some time: only perhaps coracles, for shallow freshwater marshes.